Untitled - LIT Verlag

Richard Ohnsorge
Einführung in die Mathematik

EINFÜHRUNGEN
– Naturwissenschaften –
Band 1
LIT

Richard Ohnsorge
Einführung in die Mathematik
Analysis und Lineare Algebra im Grundstudium
LIT

Bibliografische Information der Deutschen Nationalbibliothek
Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der
Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind
im Internet über http://dnb.d-nb.de abrufbar.
ISBN 978-3-643-11228-6
© LIT VERLAG Dr. W. Hopf Berlin 2011
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Auslieferung:
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Österreich: Medienlogistik Pichler-ÖBZ, e-Mail: mlo@medien-logistik.at

R n

R
Q
Z
R
R
N
Q
R

C
C
R
C
R
f : R n → R m
C

C
C
C
f : V → V
f : R n → R

L 1
L 2

N
1 1
1+1 2
1+1+1 3
1+1+1+1 4
N = {1, 2, 3,...}
1 ∈ N
R
n ∈ N n +1∈ N
m, n ∈ N m + n m · n
+:N × N → N, (1 + ...+1,
1+...+1)
↦→ 1+...+1+1+...+1
� �� � � �� � � �� � � �� �
m−mal n−mal
m−mal
1 ···
n−mal
⎫
1
⎪⎬
· : N × N → N, (1 + ...+1,
1+...+1)
↦→
� �� � � �� �
m−mal n−mal
1
�
···
⎪⎭
1
�� �
k, m, n ∈ N
m + n = n + m

k +(n + m) =(k + m)+n
1 · n = n · 1=n
m · n = n · m
k · (m · n) =(k · m) · n
(m + n)k = mk + nk
m · n =
m + n = 1+...+1
� �� �
m−mal
= 1+...+1
� �� �
n−mal
+1+...+1
� �� �
n−mal
+1+...+1
� �� �
m−mal
= n + m
k +(m + n) = 1+...+1+1+...+1+1+...+1
� �� � � �� � � �� �
k−mal m−mal n−mal
= (k + m)+n
1 ··· 1
+1
+1
⎫
⎪⎬
⎫
⎪⎬
⎪⎭
⎪⎭
1 ··· 1
� �� �
=
=+1...+1
� �� �
1 ··· ··· 1
⎫
⎪⎬
1 ··· ··· 1
� �� �⎪⎭
= n · m

1 ··· 1
1 ··· 1
� �� �
1 ··· 1
⎫
⎪⎬
1 ··· 1
� �� �⎪⎭
N
=
1 ··· 1 1 ··· 1
⎫
⎪⎬
1 ··· 1 1 ··· 1
� �� �⎪⎭
m, n ∈ N n>m k ∈ N
n>m
m am
m

a) m ∈ N
b) n ∈ N n +1∈ N
n ≥ m
n>m k ∈ N n = m + k

∀ :=
∃ :=
Z
−1 −1
−1+(−1) −2
(−1) + (−1) + (−1) −3
(−1) + ...+(−1)
� �� �
−n
N0 = N ∪{0} = {0, 1, 2, 3,...,}
Z = {...,−2, −1, 0, 1, 2,...}
n +(−m) n − m
m, n ∈ Z m + n m · n
+:Z × Z → Z, ((±1) + ...+(±1) , (±1) + ...+(±1) )
� �� � � �� �
m−mal
n−mal
↦→ (±1) + ...+(±1) +(±1) + ...+(±1)
� �� � � �� �
m−mal
n−mal
· : Z × Z → Z, ((±1) + ...+(±1) , (±1) + ...+(±1) )
� �� � � �� �
m−mal
n−mal
⎫
(±1)(±1) ··· (±1)(±1)
⎪⎬
↦→
⎪⎭
(±1)(±1) ··· (±1)(±1)
� �� �

k, m, n ∈ Z
m + n = n + m
k +(n + m) =(k + m)+n
m · n = n · m
k · (m · n) =(k · m) · n
(m + n)k = mk + nk
1+(−1) = 0 = −1+1
n +(−n) = 0 = −n + n
n +0 = n =0+n
0 · n = 0 = n · 0
1 · m = m · 1=m
1+(−1) = (−1) + 1
m + n = (±1) + ...+(±1)
� �� �
m−mal
= (∓1) + ...+(∓1)
� �� �
n−mal
±1
+(∓1) + ...+(∓1)
� �� �
n−mal
+(±1) + ...+(±1)
� �� �
m−mal
= n + m
k +(m + n) = (±1) + ...+(±1) +(±1) + ...+(±1)
� �� � � �� �
k−mal
+(±1) + ...+(±1)
� �� �
n−mal
m−mal
= (k + m)+n
(∓1)(±1) = (±1)(∓1)

m · n =
m, n
km + kn =
=
=
(±1)(∓1) ··· (±1)(∓1)
⎫
⎪⎬
⎪⎭
(±1)(∓1) ··· (±1)(∓1)
� �� �
(∓1)(±1) ··· ··· (∓1)(±1)
⎫
⎪⎬
(∓1)(±1) ··· ··· (∓1)(±1)
� �� �⎪⎭
= n · m
n>m n = m + j
((∓1)(±1)) (∓1) = (∓1) ( (±1)(∓1) )
(±1)(∓1) ··· (±1)(∓1)
(±1)(∓1) ··· (±1)(∓1)
� �� �
(±1)(∓1) ··· (±1)(∓1)
(±1)(∓1) ··· (±1)(∓1)
� �� �⎪⎭
1+−1 =0
··· (±1)(∓1)
⎫
⎪⎬
··· (±1)(∓1)
� �� �⎪⎭
⎫
⎪⎬
= k(m + n)

=
=
m>n
kn + k(−m)
(±1)1 ··· (±1)1
(±1)1 ··· (±1)1
� �� �
(±1) ··· (±1)
⎫
⎪⎬
(±1) ··· (±1)
� �� �⎪⎭
(±1)(−1) ··· (±1)(−1)
⎫
⎪⎬
(±1)(−1) ··· (±1)(−1)
� �� �⎪⎭
= kj = k(n + −(m))
k, m, n ∈ Z
m + n = n + m
k +(m + n) =(k + m)+n
∃0 ∈ Z :0+m = m +0=m
∀m ∈ Z ∃−m ∈ Z : m +(−m) =0
mn = nm
(mn)k = m(nk)
k(m + n) =km + kn
∃1 ∈ Z ∀m ∈ Z :1· m = m · 1=m
0 · n = n · 0=n
(−m)(−n) =mn
0=0+0 ′ =0 ′
0 · n = (0+0)· n =0· n +0· n
n · 0 = n · (0 + 0) = n · 0+n · 0

m, n ∈ Z
mn =0 m =0 n =0
m, n ∈ N
0 · n = 0
n · 0 = 0
k = k +0=k +(n− n)
= (k + n) − n =0− n
= −n
0 = (−m + m)(−n)
= (−m)(−n)+m(−n)
= (−m)(−n) − (mn)
(−m)(−n) =mn
mn �= 0
(−m) · n = −(mn) �= 0
m · (−n) = −(mn) �= 0
(−m) · (−n) = mn �= 0
m · n =0 m =0 n =0

P ⇒ Q
P ⇒ Q
P Q P Q
P Q P Q
P Q P ⇒ Q
¬(¬Q) = Q
¬(∃Q : Q) = ∀Q : ¬Q
¬(∀Q : Q) = ∃Q : ¬Q
¬(P ⇒ Q) = (P ¬Q)
P Q P ⇒ Q
¬

P ¬Q
¬(A ⇒ B)
P Q ¬Q P ¬Q
A ⇒ B
P ⇒ Q
A ¬A
B ¬B
¬(A ⇒ B)
¬(A ⇒ B) ⇐⇒ A ¬B
A ¬B
A ⇒ B

N0
∃c ∈ N0 : ac = b
a|b
p ∈ N,p>1 ⇐⇒
∀a ∈ N 1|a
∀a ∈ N a|0
a|b b|d a|d
a|b b �= 0 a ≤ b
bb ′ ± dd ′ ∈ N
a|b a � |d b ± d ∈ N
c =0
c = a
a|bb ′ ± dd ′
a � |b ± d
p|q p = q
1 · a = a
a · 0=0
∃c1,c2 ∈ N ac1 = b bc2 = c
c = c1c2
1 ≤ c
c =0
c ∈ N0
ac1c2 = bc2 = d
����
∈N
ac = b
⇐⇒
0=a · 0=a · c = b �= 0
a = a · 1 ≤ ac = b

∃c1,c2 ∈ N ac1 = b ac2 = d
bb ′ ± dd ′ = ac1b ′ ± ac2d ′ = a(c1b ′ ± c2d ′ )
c = c1b ′ ± c2d ′
bb ′ ± dd ′ > 0 a>0 c1b ′ ± c2d ′ > 0
∃c1 ∈ N : ac1 = b
a � |d d �= 0 d>0
a � |b − d
∃c2 ∈ N : ac2 = b − d>0
d = b − ac2
= ac1 − ac2
= a(c1 − c2)
c1 − c2 =0 d =0 d>0
c1 − c2 < 0 d0
a � |d
a � |b + d
∃c2 : ac2 = b + d
d = ac2 − b
= ac2 − ac1
= a(c2 − c1)
c2 − c1 =0 d =0 d>0
c2 − c1 < 0 d0
a � |d
q q
p|q
p = q p =1
n ∈ N,n>1
n =2 a|2 ⇒ a ≤ 2
n − 1 → n
∀2 ≤ i ≤ n − 1 i � |n
∃2 ≤ i ≤ n − 1 i|n
i1
i1|n
p = q

i1
∃n2 ∈ N
n2

a = p p|a
a>p
p|p p � |a
p � |(a − p) p � |b
a

p1
r − 1 → r
p1 ...,pr,q1 ...,qs
p1 ...pr = q1 ...qs
r = s qi
r =1
pr �= 0
s =1 p1 = q1
i ∈ 1,...,s
r − 1
p � |1 p � |p1 ...pn
pi = qi
p1 = q1 ...qs
p1
pr|p1 ...pr = q1 ...qs
pr = qs
pr|qi
p1 ...pr = q1 ...qs−1pr
pr(p1 ...pr−1 − q1 ...qs−1) =0
p1 ...pr−1 = q1 ...qs−1
r − 1=s − 1 pi = qi 1 ≤ i ≤ r − 1
p1 · ...· pn +1
∀1 ≤ i ≤ n : p �= pi
p1,...,pn

n ∈ Z\{0}
0 · 1
0
0 · 1
0
Q
n =0
= 0
= 1
n1n2 · 1
(n1n2) ·
n ∈ N −n
n +(−n) =0
1
1
n
n1 n2
1
n1n2
1
n1 n2
m
n
n · 1
n =1
1
1
1
= n1 · n2
n1 n2
= 1
=
1
n1n2
1 1
:= m =
n n m
n1 · 1
n1
n1 · n2
n1n2
1
n1
m
n1
=
= 1
= 1
n2
n1n2
= mn2
n1n2
=1
1
0

a)
b)
c)
m1
n1
m1
n1
+ m2
n2
· m2
n2
Q =
m1
n1
= m1n2
n1n2
+ m2n1
n1n2
1
1
:= m1m2
n1 n2
Q
m
n
� m
n
= m2
n2
m1 m1
=Q
n1 n1
m1 m2
=Q
n1 n2
m1 m2
=Q
n1 n2
m1n1 = m1n1
m1n2 = m2n1
= m1n2 + m2n1
n1n2
= m1m2
n1n2
Z
�
: m ∈ Z,n∈ Z\{0}
⇐⇒ m1n2 = m2n1
Z
m2 m1
=Q
n2 n1
m2 m3
=Q
n2 n3
m1n2 = m2n1
m2n3 = m3n2
m2n1 = m1n2
m1 m3
=Q
n1 n3
(m1n2)n3 = (m2n1)n3 = n1(m2n3) = n1(m3n2)
n2(m1n3 − n1m3) =0

n2 �= 0
m1
+
n1
m2
n2
m1
·
n1
m2
n2
m1
n1
m1n3 − n1m3 =0
m1n3 = n1m3
m1
=
n1
m3
n3
= s1
t1
m1n2 + m2n1
n1n2
:= m1n2 + m2n1
n1n2
:= m1m2
n1n2
m2
n2
= s2
t2
= s1t2 + s2t1
t1t2
m1t1 = n1s1
m2t2 = n2s2
n1n2(s1t2 + s2t1) = n1n2s1t2 + n1n2s2t1
m1n2 + m2n1
m1
n1
n1n2
= s1
t1
m1m2
n1n2
nisi
= n2m1t1t2 + n1m2t2t1
= t1t2(m1n2 + n1m2)
= s1t2 + s2t1
t1t2
m2
n2
= s1s2
t1t2
= s2
t2

Q
m1m2t1t2
m
n
m 1
n 1
+ 0
1
m1m2
n1n2
0:= 0
1
1:= 1
1
m1t1
= m2t2n1s1
m2t2
= n2s2n1s1
= n1n2s1s2
= s1s2
t1t2
m
n
0 ∈ Q 1 ∈
∈ Q
Def
=
m · 1 m
=
n · 1 n
Def
=
m · 1+0· n
=
n · 1
m
n
0 �= s ∈ N m, n ∈ N
0
s
=
0
1
1
1
m · s
n · s
−m
−n
=
=
=
s
s
m
n
m
n
m
−n
=
−m
n
0s = 0 = 01
1s = s = s1
mns = mns
(−m)n = m(−n)
mn = (−m)(−n)

m, n ∈ N
a ∈ Q
p1,...,pr,q1,...,qs
q ∈ Q
a = ±p1 · ...· pr
q1 · ...· qs
a = m
n
a = −m
n
m = p1 · ...· pi
n = q1 · ...· qj
a = ±p1 · ...· pr
q1 · ...· qs
x, y, z ∈ Q
x +(y + z) =(x + y)+z
x + y = y + x
x +0=x.
−x ∈ Q x +(−x) =0
(xy)z = x(yz)
xy = yx
1 �= 0 x · 1=x
x −1 ∈ Q xx −1 =1
x(y + z) =xy + xz

m
n
∈ Q
−m
m1
n1
= m1
n1
�
m2
+ +
n2
m3
�
n3
+ m2n3 + m3n2
n2n3
= m1n2n3 + n1m2n3 + n1m3n2
n1n2n3
= m1n2n3 + m2n1n3 + n2n1m3
n1n2n3
= m1n2 + m2n1
=
m1
n1
n
m
n
� m1
m1
n1
n1
� m1
n1
+ m2
n2
∈ Q
+ −m
n
· m2
n2
n1n2
· m2
�
m3
n2 n3
+ m2
n2
+ m3
n3
�
+ m3
n3
= m1n2 + m2n1
n1n2
= m2n1 + m1n2
n1n2
= m2
n2
= mn − nm
nn
= m1m2
n1n2
+ m1
n1
= m1m2
n1n2
= 0 0
=
nn 1
· m3
n3
= m1m2m3
n1n2n3
= m1
� �
m2m3
n1 n2n3
= m1
�
m2
·
n1 n2
m3
n3
= m2m1
n2n1
= m2
n2
�
· m1
n1

x · 1=x
1 0
1 = 1
0 �= m
n
∈ Q
n
0=1· 0=1· 1=1
m ∈ Q
m
n
n
m
�
m1 m2
n1
mn 1
= =
nm 1
n2
+ m3
�
n3
= m1 m2n3 + n2m3
n1 n2n3
= m1m2n3 + m1n2m3
n1n2n3
= n1 (m1m2n3 + n2m1m3)
n1 (n1n2n3)
= m1m2
n1n2
+ m3
n3
= m1 m2
+
n1 n2
m1 m3
n1 n3

x 0
< 0
= 0
q1,q2 ∈ Q q1 = m1
n1 > 0 q2 = m2
> 0 n2
q1 + q2 > 0
q1 · q2 > 0
q1 > 0 q2 > 0 m1,m2 > 0
m1
+
n1
m2
n2
m1 m2
n1 n2
m1n2 + m2n1 > 0
m1m2 > 0
= m1n2 + m2n1
n1n2
= m1m2
n1n2
> 0
> 0
x, y ∈ Q
x>y ⇐⇒ x − y>0
x ≥ y ⇐⇒ x − y>0 x − y =0
x0
x ≤ y ⇐⇒ y − x>0 y − x =0
x

x0
a + y − (a + x) = y − x>0
ay − ax =
����
a · (y − x) > 0
� �� �
>0 >0
y − x>0 y ′ − x ′ > 0
y + y ′ − (x + x ′ )=y − x
y − x>0 b − a>0
� �� �
>0
x + x ′ 0
by − ax = by − bx + bx − ax
= b · (y − x)+(b− a) ·
� �� � ����
x
>0 ≥0
≥
����
b · (y − x)
� �� �
>0 >0
> 0
x0
x

x0
0 �= x ∈ Q
0 0
x · x>0
x −1 > 0
x −1 < 0
x −1 >y −1
< 0 ⇒ −−m
n
> 0 ⇒ −m>0
⇒
⇒
m0 y>0
x>0: x
����
>0
� �� �
>0
·
����
x > 0
>0
x0: x −1 = x
����
>0
� �� �
>0
x0
� �� �
>0
· y > 0
����
>0
= m
n
> 0
· (y − x) > 0
� �� �
>0
· (−x) > 0
� �� �
>0
· x −1 x −1
� �� �
> 0
>0
· x −1 x −1
� �� �
>0
> 0

xyy −1 x −1 =1
(xy) −1 = x −1
����
>0
y −1
> 0
����
>0
x −1 − y −1 = x −1 yy −1 − y −1 xx −1
= x −1 y −1
> 0
� �� �
>0
· (y − x)
� �� �
>0
a ∈ Q a>0 n ∈ N
n y

x = p1 ...pr
q1 ...qs
y = a1 ...at
b1 ...bu
n − 1=q1 ...qsa1 ...at
(n − 1)x − y = q1 ...qsa1 ...atp1 ...pr
q1 ...qs
− a1 ...at
b1 ...bu
= q1 ...qsa1 ...at(p1 ...prb1 ...bu − 1)
q1 ...qsb1 ...bu
≥ 0
nx − y = x +(n− 1)x − y ≥ x>0
� �� �
≥0
x 0
x 1
x n
x −n
:= 1
:= x
:= x
�
· ...·
��
x
�
:= x −1 · ...· x −1
� �� �

x0
|q| := 0
⎩
−q
q =0
q0: |x| = x>0
x =0: |x| =0
x0
|x| = x = −(−x) =|−x|
|x| = −x = |−x|
����
>0

x ≥ 0,y ≥ 0: |x||y| = xy = |xy|
x0
y
����
≥0
= − xy = |xy|
����
≤0
x0 >0
x ≥ 0,y 0 ≤0
x = x
y y
� �
�
|x| = �
x�
�
� y � |y|
|x|
|y| =
� �
�
�
x�
�
� y �
x ≥ 0: |x|−x = x − x =0
x 0
� �� �
>0 >0
x ≥ 0: |x| + x = x + x = 2 x > 0
����
>0
x0

|y| + |x|−|x + y| ≥0
|x| = |x + y +(−y)|
≤ |x + y| + |−y|
= |x + y| + |y|
−|y|
|x|−|y| ≤|x + y|

Q R
R
(a, b) := {x ∈ R : a

x ∈
x ∈
� �
i∈I
� �
i∈I
⇐⇒
x ∈ A C ⇐⇒ x �∈ A
� A C �C = A
A ⊂ B ⇒ B C ⊂ A C
� �C �
= �
i∈I
� �
i∈I
Ai
Ai
� C
i∈I
= �
i∈I
A C i
A C i
�
(Ai ∩ B) = B ∩ �
i∈I
i∈I
Ai
x ∈ � A C�C ⇐⇒ x �∈ A C
⇐⇒ x ∈ A
x ∈ B C ⇐⇒ x �∈ B
Ai
Ai
� C
� C
A⊂B
⇒ x �∈ A
⇐⇒ x ∈ A C
⇐⇒ x �∈ �
i∈I
Ai
⇐⇒ ¬ (∃i ∈ I : x ∈ Ai)
⇐⇒
⇐⇒
∀i ∈ I : x �∈ Ai
x ∈ �
i∈I
⇐⇒ x �∈ �
i∈I
A C i
Ai
⇐⇒ ¬ (∀i ∈ I : x ∈ Ai)
⇐⇒
⇐⇒
∃i ∈ I : x �∈ Ai
x ∈ �
i∈I
A C i

w ∈ �
(Ai ∩ B) ⇐⇒ ∀i ∈ I : w ∈ Ai ∩ B
i∈I
⇐⇒ ∀i ∈ I : w ∈ Ai w ∈ B
⇐⇒ w ∈ B ∩ �
i∈I
Ai

(an)n
an = 1
n
n ∈ N
an =(−1) n n ∈ N
an =1 n ∈ N
⇐⇒ an a
⇐⇒
an
(1, 1, 1,...)
�
1, 1 1 1
, ,
2 3 4 ,...
�
(−1, 1, −1, 1,...)
a
ε
an ∈ R (a1,a2,...)
ε>0
⇐⇒ ε>0 n0 −ε ≤ an − a ≤ ε
(an)n
∀ε >0 ∃n0 ∈ N ∀n ≥ n0 : |an − a| N
limn→∞ an = ∞
(an)n
−∞ ⇐⇒
∀N ∈ N ∃n0 ∈ N ∀n ≥ n0 : an < −N
limn→∞ an = −∞
⇐⇒
∃K ∈ N ∀n ∈ N : |an| ≤K
a ∈ R ⇐⇒
∞ ⇐⇒

limn→∞ an = a
∀ε >0:|a − a ′ |≤ε a = a ′
ε = |a−a′ |
2
a ′
n1
n2
ε =1
a �= a ′ |a − a ′ | > 0
ε = |a − a′ |
2
< |a − a ′ |≤ε
n ≥ n1 : |an − a| < ε
2
n ≥ n2 : |an − a ′ | < ε
2
n ≥ max{n1,n2}
a = a ′
|a ′ − a| ≤ |a ′ − an| + |an − a|
< ε ε
+ = ε
2 2
∃n0 ∀n ≥ n0 : |an − a| ≤1
∀n ≥ n0 : |an| = |an − a + a|
≤ |an−a| + |a| ≤|a| +1
K =max{|a1|,...,|an0|, |a| +1}
∀n ∈ N : |an| ≤K
limn→∞ an = a limn→∞ bn = b c ∈ R
limn→∞(an + bn) = limn→∞ an + limn→∞ bn
limn→∞ can = c limn→∞ an
limn→∞(anbn) = (limn→∞ an)limn→∞ bn
b �= 0
an
lim =
n→∞ bn
limn→∞ an
limn→∞ bn

∃n1 ∀n ≥ n1 : |an − a| < ε
2
∃n2 ∀n ≥ n2 : |bn − b| < ε
2
n0 := max{n1,n2} n ≥ n0
c �= 0
(an)n
c =0
(an)n
(bn)n
∀n ≥ n1
|an + bn − (a + b)| ≤ |an−a| + |bn − b|
< ε ε
+ = ε
2 2
(bn)n
lim
n→∞ can = lim
n→∞ 0=0
∃n1 ∀n ≥ n1 : |an − a| < ε
|c|
|can − ca| = |c||an − a|
< |c| ε
= ε
|c|
∃n1 ∀n ≥ n1 : |an − a| < ε
2|b|
∃n1 ∀n ≥ n2 : |bn − b| < ε
2K
n ≥ max(n1,n2)
|anbn − ab| = |anbn − anb + anb − ab|
≤
<
|an||bn − b| + |b||an − a|
|an| ε ε
+ |b|
2K 2|b|
≤ ε ε
+ = ε
2 2
∃n1 ∀n ≥ n1 : |bn − b| < |b|
2
K>0

(an)n
n ≥ n1
|bn| = |bn − b + b| ≥|b|−|bn − b| >
� �� �
0
2
(bn)n
bn
∃n2 ∀n ≥ n2 : |an − a| < ε|b|
4
∃n3 ∀n ≥ n3 : |bn − b| < ε|b|2
4|a|
n0 := max{n1,n2,n3} n ≥ n0
�
�an
�
� −
bn
a
�
�
�
b � =
� �
�
�
anb − bna �
�
� bbn
�
= |anb − ab + ba − bna|
|bbn|
≤ |b||an − a| + |b − bn||a|
|bbn|
< 2|an − a|
+
|b|
2|a|
|b| 2 |bn − b|
< 2 ε|b| 2|a|
+
|b| 4 |b| 2
|b| 2ε 4|a|
= ε ε
+ = ε
2 2
limn→∞ an = a limn→∞ bn = b
∀n ∈ N : an ≤ bn
∀n ∈ N : an

lim
n→∞ (bn − an) = lim
n→∞ bn − lim
n→∞ an = b − a
∃n0 ∀n ≥ n0 : |bn − an − (b − a)| <
b − a0 N> 1
ε
=0
= ∞
∃n0 ∀n ≥ n0 : |an| >N
|an| > 0 n ≥ n0
� �
�
∃n0 ∀n ≥ n0 : �
1 �
� − 0�
� =
� �
�
�
1 �
�
� �
an
an
(B)n
< 1
N

ε = 1
N
anbn
n0
∀N ∈ N ∃n0 ∀n ≥ n0 : |an − 0| = N
ε
∀ε >0 ∃n0 ∀n, m ≥ n0 : |an − am|

ε =1
R
∃n0 ∀n, m ≥ n0 : |an − am| < 1
∀n ≥ n0 : |an0 − an| < 1
K := max{|a1|,...,|an0 | +1}
x ≥−1
∀c ∈ R+ ∃k ∈ N :2 k ≥ c
n → n +1:
k ≥ c
n =1
(1 + x) n+1
∀n ∈ N : |an| ≤K
(1 + x) n ≥ 1+nx
(1 + x) 1 =1+x
R Q
= (1+x) n (1 + x)
1+x≥0
≥ (1 + nx)(1 + x)
= 1+(n +1)x + nx 2
≥ 1+(n +1)x
2 k = (1+1) k ≥ 1+k
≥ k ≥ c
(ank )k
{an : n ∈ N}
(an)n
(an)n

M0 = A0+B0
2
an0
[A0,M0]
[M0,B0]
Mk = Ak+Bk
2
(an)n
K>0 −K ≤ an ≤ K
A0 = −K
B0 = K
an
A1 := A0
B1 := M0
A1 := M0
B1 := B0
R
[A0,M0], [M0,B0]
[Ak,Mk], [Mk,Bk]
an ank nk >
nk−1 Ak+1,Bk+1 k ∈ N :
(ank )k
i, j ≥ k
k> ε
2K
[Ak,Bk] ⊂ [Ak−1,Bk−1]
Bk − Ak = 2K
2 k
∀i ≥ k : Ak ≤ ani
Ak ≤ ani,anj ≤ Bk
≤ Bk
|ani − anj |≤Bk − Ak = 2K
2 k
2 k >k+1>k≥ ε
2K
|ani
− anj |≤2K

∀n ∈ N : a2n =1
∀n ∈ N : a2n+1 = −1
c �∈ {−1, 1}
(ank )k limk→∞ ank
an =(−1) n
= a
lim
n→∞ a2n = lim
n→∞ 1=1
lim
n→∞ a2n+1 = lim
n→∞ −1=−1
|c − an| ≥min{|c − 1|, |c +1|} > 0
|c − an| +1, −1
a =limn→∞ an
(an)n
(an)n ⇐⇒
(an)n limk→∞ ank
∀ε >0 ∃n0 ∀n ≥ n0 : |an − a| ≤ε
ank limk→∞ ank
(an)n
a (an)n a
ε>0 (an)n
(ank )k
b = a
∀ε >0 ∃N ∈ N ∀n, m ≥ N : |an − am| < ε
2
a
∀ε >0 ∃K ∈ N ∀k ≥ K : |ank
∀n ≥ max(N,K)
− a| < ε
2
ε ε
|an − a| ≤|an− ank | + |ank − a| < + = ε
2 2
= b
= a
(ank )k

a
⇐⇒ an+1 >an
⇐⇒ an+1 ≤ an
⇐⇒ an+1

(xn)n
(Kn)n
Mn y ∈ A y>Mn
Kn+1 = Kn
xn+1 := y
xn ≤ xn+1
Kn+1 ≤ Kn
x0 ≤ xn ≤ xn+1 ≤ Kn+1 ≤ Kn ≤ K0
Kn − xn ≤ 2 −n (K0 − x0)
x0
K0
x K R
∀n ∈ N : xn ≤ xn+1 ≤ Kn+1 ≤ Kn
xn ≤ x ≤ Kn
xn ≤ K ≤ Kn
xn ≤ x ≤ K ≤ Kn
ε>0 n> K0−x0
ε
|x − K| ≤ Kn − xn
≤ 1
2n (K0 − x0) ≤ 1
n (K0
<
− x0)
ε
x = K
lim inf
n→∞ an = sup inf
lim sup
n→∞
n
m≥n am =
an = inf sup am =
n m≥n
lim infn→∞ an =limsup n→∞ an ∈ R
lim
n→∞ an
lim inf
n→∞ an =limsup=
lim
n→∞ n→∞ an

(infm≥n am)n
{am : m ≥ n +1}⊂{am : m ≥ n}
inf
m≥n+1 am ≥ inf
m≥n am
lim
n→∞ inf
m≥n+1 am =supinf
n∈N m≥n am
lim infn→∞ an ∈ R
∀ε >0 ∃N ∀n ≥ N :
�
�
�
� inf
m≥n am − lim inf
n→∞ an
�
�
�
� <
ε
2
∀n ∈ N : inf
m≥n am
�
�
∀n ∈ N ∀ε >0 ∃nk >n: �ank
� − inf
m≥n
�= ±∞
am
�
�
�
� <
ε
2
(ank )k
ε = 1
k
Nk ≥ nk−1 nk ≥ Nk
�
�
�ank − lim inf
n→∞ an
�
�
� ≤
�
�
�ank
� − inf
< 1 1 1
+ =
2k 2k k
lim ank
k→∞
= liminf
n→∞ an
lim infn→∞ an
(ank )k
lim infn→∞ an
lim infn→∞ an = −∞
∀nk : ank
am
m≥nk
an
�
�
�
� +
�
�
�
� inf am − lim inf
m≥nk
(an)n
≥ inf am
m≥nk
c = lim ank ≥ lim
k→∞ n→∞ inf
m≥n am
∀n ∈ N : inf
m≥n am = −∞
∀n, K ∈ N ∃nk >n: ank < −K
n→∞ an
�
�
�
�
±∞

nk >nk−1
lim infn→∞ an
nk >nk−1
lim infn→∞ an
nk >nk−1
lim infn→∞ an
lim sup
n→∞
lim ank = −∞
k→∞
lim inf
n→∞ an = ∞
∀n ∈ N : inf
m≥n am < ∞
(ank )k
∀K ∈ N ∃n ∈ N : inf
m≥n am ≥ K
∀K, n ∈ N ∃nk >n: ank ≥ K
lim ank = ∞
k→∞
lim inf
n→∞ an = ∞
∀n ∈ N : inf
m≥n am = ∞
∀K, n ∈ N ∃nk >n: ank
an =infsup
am =
n m≥n
lim ank = ∞
k→∞
(ank )k
≥ K
∀n ∈ N : inf
m≥n am ≤ an ≤ sup am
m≥n
�
�
∃n0 ∀n ≥ n0 : �
� inf
m≥n am − lim inf
�
�
∃n0 ∀n ≥ n0 : �
� sup am − lim sup
m≥n n→∞
n→∞ an
an
�
�
�
�

∀n ≥ n0
ε>0
−ε + lim inf
n→∞ an ≤ inf
m≥n am
lim
n→∞ an = lim inf
≤ an
≤ sup am
m≥n
≤ lim sup an + ε
n→∞
= liminf
n→∞ an + ε
n→∞ an =limsupan
n→∞

(an)n
(an)n
(an ± bn)n
(anbn)n
(bn)n
( an
bn )n
(an)n
Q R
∃ε >0 ∃n0 ∈ N ∀n ≥ n0 : an >ε
∃ε >0 ∃n0 ∈ N ∀n ≥ n0 : an < −ε
R
∃ε >0 ∀n1 ∈ N ∃n0 ≥ n1 : |an0 − 0| > 2ε
∃ε >0 ∀n1 ∈ N ∃n0 ≥ n1 :(an0 > 2ε an0 < −2ε)
∃n1 ∈ N ∀n, m ≥ n1 : |an − am| 0 n1 ∈ N
n0 ∈ N ∀n ≥ n0 :
(an)n
(an)n, (bn)n
+ an0
an = an − an0
> −|an − an0 | +2ε ≥ ε
+ an0
an = an − an0
< |an − an0 |−2ε

(bn)n
(an)n
m, n ≥ max(n1,n2)
|an ± bn − (am ± bm)| ≤ |an−am| + |bn − bm|
< ε ε
+ = ε
2 2
(an)n, (bn)n
(an)n
∃n1 ∀m, n ≥ n1 : |an − am| < ε
2K
∃n2 ∀m, n ≥ n2 : |bn − bm| < ε
2K
m, n ≥ max(n1,n2)
|anbn − ambm| = |anbn − anbm + anbm − ambm|
≤
<
|an||bn − bm| + |bm||an − am|
K ε ε
+ K = ε
2K 2K
(bn)n
∃δ >0 ∃n0 ∀n >n0 : |bn| >δ
∃n2 ∀m, n ≥ n2 : |an − am| < εδ
2
∃n3 ∀m, n ≥ n3 : |bn − bm| < εδ2
2K
K>0
n0 := max{n1,n2,n3} m, n ≥ n0
�
�an
�
� −
bn
am
�
�
�
bm
� =
�
�
�anbm
�
− bnam �
�
� bmbn
�
= |anbm − ambm + bmam − bnam|
|bmbn|
≤ |bm||an − am| + |bm − bn||am|
|bmbn|
< 1
δ |an − am| + K
δ2 |bn − bm|
≤ 1 εδ K
+
δ 2 δ2 εδ2 =
2K
ε ε
+ = ε
2 2
K>0

(an)n, (bn)n an bn
(bn)n
R := {(an)n :(an)n
⇐⇒
,an ∈ Q}
(an)n ∼ (bn)n ⇐⇒ lim
n→∞ (an − bn) =0
lim
n→∞ (an − an) =0
(an)n ∼ (an)n
lim
n→∞ (an − bn) =0⇒ lim
n→∞ (bn − an) =0
(an)n ∼ (bn)n ⇒ (bn)n ∼ (an)n
lim
n→∞ (an − bn) =0 lim
n→∞ (bn − cn) =0
⇒ lim
n→∞ (an − cn) = lim
n→∞ (an − bn) + lim
n→∞ (bn − cn) =0
(an)n ∼ (bn)n
(bn)n ∼ (cn)n ⇒ (an)n ∼ (cn)n
R
(an)n ± (bn)n := (an ± bn)n
(an)n · (bn)n := (an · bn)n
(an)n
:=
(bn)n
� an
bn
�
n

ε
an
bn
n ≥ n0
∃n0 ∀n ≥ n0 : |bn| >ε
(an)n, (bn)n, (cn)n (dn)n ∀n ≥ n0 : |bn|, |dn| >
K>0
limn→∞(an − cn) =0 limn→∞(bn − dn) =0
0 ≤ lim
n→∞
�
�
�
�
lim
n→∞ (an ± bn − (cn ± dn))
= lim
n→∞ (an − cn) ± lim
n→∞ (bn − dn)
= 0
an ∼ cn,bn ∼ dn ⇒ an ± bn ∼ cn ± dn
0 ≤ lim
n→∞ |anbn − cndn|
≤ lim
n→∞ |anbn − andn + andn − cndn|
= K lim
n→∞ |bn − dn| + K lim
n→∞ |an − cn|
= 0
an
−
bn
cn
dn
an ∼ cn,bn ∼ dn ⇒ anbn ∼ cndn
�
�
�
�
= lim
�
�
�
�
andn − cnbn
�
�
�
�
n→∞ bndn
≤ 1
lim
ε2 n→∞ |andn − anbn + anbn − cnbn|
≤ K limn→∞ |dn − bn| + K limn→∞ |an − cn|
ε2 = 0
an ∼ cn,bn ∼ dn ⇒ an
bn
∼ cn
dn

x, y, z ∈ R
x +(y + z) =(x + y)+z
x + y = y + x
x +0=x =0+x
−x ∈ R x +(−x) =0
(xy)z = x(yz)
xy = yx
∃1 �= 0: x · 1=x
∀0 �= x ∈ R ∃x −1 ∈ R xx −1 =1
x(y + z) =xy + xz
(an)n
1
an
∃ε >0 ∃n0 ∀n ≥ n0 : |an| >ε
n ≥ n0
an +(bn + cn) = (an + bn)+cn
an + bn = bn + an
an +0 = an = an +0
(anbn)cn = an(bncn)
anbn = bnan
an · 1 = 1· an = an
1
= 1 = 1
an
an
an
an
an(bn + cn) = anbn + ancn
(an)n
an ∈ Q
(an)n = 0 lim
n→∞ an =0
(an)n > 0 ∃N0 ∈ N ∃ε >0 ∀n ≥ N0 : an >ε
(an)n < 0 ∃N0 ∈ N ∃ε >0 ∀n ≥ N0 : an < −ε
(an)n
(an)n > 0 (bn)n > 0
(an)n +(bn)n > 0
(an)n · (bn)n > 0

∃ε1,n1 ∀n ≥ n1 : an >ε1
∃ε2,n2 ∀n ≥ n2 : bn >ε2
n3 := max(n1,n2) ε := min(ε1 + ε2,ε1ε2) ∀n ≥ n3
(an)n
>, ε1 + ε2 ≥ ε
an · bn > ε1 · ε2 ≥ ε
⎧
⎨ a
a ∈ R
a>0
|a| := 0
⎩
−a
a =0
a0 ∃n0 ∈ N ∀n ≥ n0 : |an| 0 ∃n0 ∈ N ∀n ≥ n0 : |an| 0 ∃k0 ∀k ≥ k0 : |a k − a| 0 ∃k0 ∀k, l ≥ k0 : |a k − a l |

n0
((a k n)n)k
∀ε >0 ∃k0 ∀k, l ≥ k0 ∃ε >δ>0 ∃n0 ∀n ≥ n0 : |a k n − a l n| < ε
− δ
2
R
(bn)n ∈ R
∀n ∈ N : bn := a n n
n, m ≥ k (a m n )n Q
(bn)n
(bn)n
∃n1 >n0 ∀m, n ≥ n1 : |a m n − a m m| < ε
2
∀n, m ≥ n2 := max(n1,k0)
|bn − bm| Def
= |a n n − a m m|
≤ |a n n − a m n | + |a m n − a m <
m|
ε ε
− δ + = ε − δ
2 2
Q
∀ε >0 ∃k0 ∀k ≥ k0 ∃ε >δ>0 ∃n2 ∈ N ∀n ≥ n2 :
|bn − a k n| Def
= |a n n − a k n| < ε
− δ 0
((a k n)n)k
l ≥ k0
(bn)n
∀x, y ∈ (0, ∞) ∃N ∈ N : Nx > y
∃ε >0 ∃n0 ∀n ≥ n0 : an >ε bn >ε
Q
ε
∃n1 ≥ n0 ∀n ≥ n1 : |an − an1 | <
2
|bn − bn1| < ε
2
δ
a m m

Q
�
∃N ∈ N : N
an1
∀n ≥ n1 :
�
Nan − bn ≥ N
ε
�
ε
− >bn1 + + δ
2 2
an1
> δ > 0
Nx > y
ε
�
ε
− − bn1 −
2 2

f : D ⊂ R → R
g : E ⊂ R → R f(D) ⊂ E
R ⊃ D
f ◦ g
f◦g
−→ R
f ↘ ↗ g
E ⊂ R
f ◦ g : D → R,x↦→ f(g(x))
a ∈ D (an)n
limn→∞ an = a (an)n limn→∞ an = a
lim
n→∞ f(an) =c
lim f(x) =c
x→a
lim f(x) =f(a)
x→a
⇐⇒ (xn)n limn→∞ xn = a
�
f lim
n→∞ xn
�
= lim
n→∞ f(xn)
f : D ⊂ R → R p ∈ D ⇐⇒
∀ε >0 ∃δ >0 ∀x ∈ D :
⇒
∃ε >0 ∀δ >0 ∃x ∈ D :
a ⇐⇒
|x − p|

ε>0 δ = 1
n
∀n ∈ N ∃xn ∈ D
|xn − p| < 1
n
|f(xn) − f(p)| ≥ε
(xn)n limn→∞ xn = p
limn→∞ f(xn) =f(p)
∀n ∈ N : |f(xn) − f(p)| ≥ε
⇐ ε>0 limn→∞ xn = p
∃n0 ∀n ≥ n0 : |xn − p| ≤δ
|f(xn) − f(p)| 0 ∀x ∈ D ∩ (p − δ, p + δ) :f(x) �= 0
(p − δ, p + δ)
ε = |f(p)| > 0 δ>0
|x − p|

∀ε >0 ∃n0 ∈ N ∀n ≥ n0
limn→∞ f(an) =f(a)
f(a) �= 0
|f(an) − f(a)| = |c − c| =00 ∃n0 ∈ N ∀n ≥ n0 : |f(an) − f(a)| = |an − a|

g ◦ f
f : D → R,g : E → R f(D) ⊂ E
f(a)
R ⊃ D
f◦g
−→ R
f ↘ ↗ g
E ⊂ R
(xn)n
lim
n→∞ (f ◦ g)(xn) = lim
n→∞ (f(g(xn)))
�
= f lim
n→∞ g(xn)
�
� � ��
= f g
= f(g(a))
[c0,d0] :=[a, b]
limn→∞ xn = a
lim
n→∞ xn
f :[a, b] → R f(a) < 0 f(b) > 0
p ∈ [a, b] f(p) =0
[cn,dn] ⊂ [cn−1,dn−1]
dn − cn =2 −n (b − a)
f(cn) ≤ 0
f(dn) ≥ 0
n =0
n → n +1 [cn,dn] m =(cn + dn)/2
f(m) ≥ 0 [cn+1,dn+1] =[cn,m]
f(m) < 0 [cn+1,dn+1] =[m, dn]
(cn)n
(dn)n
∀ε >0 ∃n0 ∀n ≥ n0
a ≤ cn ≤ cn+1 ≤ dn+1 ≤ dn ≤ b
cn ≤ c, d ≤ dn
|c − d| ≤|cn − dn| =2 −n (b − a)

f(c) =0
−g
c = d
0 ≤ lim
n→∞ f(cn) =f(c) =f(d)
= lim
n→∞ f(dn) ≤ 0
f :[a, b] → R
p ∈ [a, b] f(p) =c
f(a) =c f(b) =c p = a p = b
f(a) f(b)
g(x) =f(x) − c
g(a) < 0

c := sup x∈[a,b] f(x)
(xn)n
(xnk )k
a ≤ xnk ≤ b
c := inf x∈[a,b] f(x)
⇐⇒
∀ε >0 ∃δ >0 ∀x, x ′ ∈ D
∃ε >0 ∀δ >0 ∃x, x ′ ∈ D
lim
n→∞ f(xn) =c
(xn)n
[a, b]
(f(xn))n
a ≤ p = lim xnk ≤ b
k→∞
f(p) = lim f(xnk )=c
k→∞
f : D → R
|x − x ′ |

f = 1
g
δ>0
x ′ nk
lim
�
= f
k→∞
lim
k→∞ x′ nk = p
� f(x ′ nk
lim
k→∞ x′ nk
= f(p) − f(p) =0
) − f(xnk )�
� �
− f
lim
k→∞ xnk
∀k ∈ N : f(x ′ nk ) − f(xnk ) ≥ ε
f :(0, ∞) → (0, ∞),x↦→ 1/x
g : x → x x ∈ (0, ∞)
�
�
�
�f g(x) =x �= 0
� �
�
�
1 1 �
� − �
n 2n�
= 1
2n 1
n 2n
∃ ε =1> 0 ∀δ >0 ∃ 1 1
n , 2n ∈ (0, ∞)
�
�
�
�f � 1
n
�
− f
� ��
1 ���
>ε
2n
�

f : D → R
⇐⇒ ( x

f −1
y ∈ Y x := f −1 (y) ∈ X
f :[a, b] → R
f(x) =f(f −1 (y)) = y
f −1 :[f(a),f(b)] → [a, b] ⊂ R
f f(x) =f(y)
xf(y)
x = y
f p ∈ [f(a),f(b)]
f −1
c, d ∈ [f(a),f(b)] cf(y) =d
x = y c = f(x) =f(y) =d
x

(xn)n
(xnk )k limk→∞ xnk
yn = f(xn)
y = lim ynk = lim f(xnk )
k→∞ k→∞
� �
= f
= f(p)
lim
k→∞ xnk
f −1 y = f(p)
∃ε >0 ∀δ >0 ∀y ′ ∈ D
|y ′ − f(p)| ε
y ′ = ynk
= f(xnk ) ∀k ∈ N
|f(xnk ) − f(p)| ε
limk→∞ xnk
= p
k ≥ 2,k ∈ N
f :[0, ∞) → [0, ∞),x↦→ x k
f −1 :[0, ∞) → [0, ∞),x↦→ k√ x
n =2 0 ≤ x

x ∈ R n>x [0,n k ] [0, ∞)
f :[0, ∞) → [0, ∞)
f −1 :[0, ∞) → [0, ∞)

n =1
R
√ 2 �∈ Q.
√ 2 ∈ Q ∃m, n ∈ N
√ 2= m
n = p1 ...pr
q1 ...qt
∀i, j : pi �= qj
t ≥ 1 √ 2=p1 ···pr
2=p 2 1 ...p 2 r
r =0 2 �= 1
r ≥ 1 pi ≥ 2
4 ≤ p 2 1 ...p 2 r =2
2= p1 ...prp1 ...pr
q1 ...qtq1 ...qt
2q1 ...qtq1 ...qt = p1 ...prp1 ...pr
∃i : q1|pi
∃i : q1 = pi
n ∈ N xn > 0
∀i, j : qi �= pj
(xn)n Q limn→∞ xn = √ 2
(xn)n
x1 := 3
2
xn+1 := 1
2 xn + 1
xn ∈ Q
x1 = 3
> 0
2
xn

n → n +1
xn+1 := 1
2
xn
����
>0, n
+ 1
> 0
xn
����
>0
n ∈ N x2 n =1
n − 1 → n
n − 2 ≥ 0
� �2 3 9 − 8 1
− 2= = > 0
2
4 4
x 2 n − 2 =
�
xn−1
2
(xn)n
(xn)n
= x2 n−1
=
�2 1
+ − 2
xn−1
1
+1+
4
�
xn−1 1
−
2 xn−1
xn − xn+1 = xn − xn
2
=
1
2xn
(xn)n
(yn)n yn := 2
xn
∀n ∈ N : xn > 0
x2 n−1
�2 − 2
≥ 0
1
− =
xn
xn 2
−
2 2xn
(x 2 n − 2) ≥ 0
xn ≥ xn+1 > 0
yn+1 = 1
xn+1
n ∈ N y 2 n ≤ 2 ≤ x 2 n
x 2 n ≥ 2
y 2 n =
� 2
xn
� 2
≥ 1
= yn
xn
�
2
≤ 2
x2 �
≤ 2
n
� �� �
≤1
s

∀n ∈ N
n ∈ N xn ≥ yn
yn >xn > 0
yn >xn
1 2
=
3 x1
s = √ 2
limn→∞ 1 1 = xn s > 0
y1
y 2 n >x 2 n
d) f)
≤ yn ≤ xn
= y1 ≤ lim
n→∞ xn = s
s = lim
n→∞ xn+1 = lim
n→∞
= s 1
+
2 s
s
2
=
1
s
s 2 s
=
=
2
√ 2
x0 =1
f(xn) =x 2 n − 2=0
xn+1 = xn − f(xn)
f ′ (xn)
= xn − x2 n − 2
2xn
= xn
2
+ 1
xn
�
1
2 xn + 1
�
xn

N
Z
∀n ∈ N Mn
Q
N → N,n↦→ n
Z
xnj ∈ Mn
∃f : N → A : f
⇐⇒
A = {a1,...,an}
f : N → A, i ↦→
(yn)n =(x00,x10,x01
� �� �
n+j=1
�
ai
a1
�
n∈N Mn
1 ≤ i ≤ n
i>n
(yn)n =(0, −1, 1, −2, 2, −3, 3,...)
f : N → Z,n↦→ yn
⇐⇒
,x20,x11,x02,x30,x21,x12,x03
� �� �
n+j=2
�
Mn = {yn : n ∈ N}
n∈N
f : N → �
n∈N
∀n ∈ N : An =
Q = �
Mn,n↦→ yn
n∈N
� m
n
An
� �� �
n+j=3
�
: m ∈ Z
,...)

an ∈{0,...,9}
x ≥ 0
m → m +1
�
0 ≤ x −
x ∈ R
x = ±
∞�
n=−k
an10 −n
∃m ∈ N : 0 ≤ x

(0, 1)
(xn)n
xn = 0,an1 an2 an3 ...
z = 0,z1 z2 z3 zn =
∀(xn)n
(0, 1)
� ann +2 ann < 5
ann − 2 ann ≥ 5
∀n ∈ N : |zn − ann| =2
∀n ∈ N : |z − xn| ≥10 −n
∀n ∈ N : z �= xn
(0, 1) ∃z ∈ (0, 1) ∀n ∈ N : xn �= z
(0, 1) (xn)n (0, 1)

x + h ∈ D
f : D ⊂ R → R x ∈ D ⇐⇒
f ′ f(y) − f(x)
(x) := lim
y→x,y�=x y − x
yn ∈ D yn → x
f : D ⊂ R → R x ∈ D ⇐⇒
⇒ y = x + h
⇐ h = y − x
f : R → R
f ′ f(x + h) − f(x)
(x) := lim
h→0 h
f(a)+c(x − a)
f : D ⊂ R → R ⇐⇒
∃c ∈ R ∃ r : D → R
⇒
r(x) =
lim
x→a,x�=a
lim
x→a,x�=a
f(x) = f(a)+c(x − a)+r(x)
r(a) = 0
r(x)
x − a
= 0
c = f ′ (a)
� f(x) − f(a) − f ′ (a)(x − a) x �= a
0 x = a
r(x)
x − a
= lim
x→a,x�=a
f(x) − f(a)
x − a
− f ′ (a)
= f ′ (a) − f ′ (a) =0

⇐
�
f(x) − f(a)
lim
= lim c +
x→a,x�=a x − a x→a,x�=a
r(x)
�
= c
x − a
c = f ′ (a)
(xn)n limn→∞ xn = a ∀n ∈ N : xn �= a
lim
n→∞
r(xn)
xn − a =0
⇐⇒ ∃ (cn)n limn→∞ cn =0
⇐
⇒
r(xn) =cn(xn − a)
cn := r(xn)
xn − a
lim
n→∞ cn
r(xn)
= lim
n→∞ xn − a =0
lim
n→∞
f : D ⊂ R → R
r(xn)
= lim
xn − a n→∞ cn =0
limn→∞ xn = a xn ∈ D ∀n : xn �= a
lim
n→∞ f(xn) = lim
n→∞ (f(a)+c(xn − a)+r(xn))
= f(a) + 0 + lim
n→∞ cn lim
n→∞ (xn − a)
�
= f(a) =f
lim
n→∞ xn
f,g : D ⊂ R → R x ∈ D c ∈ R
f + g, f · g cf
(f + g) ′ (x) =f ′ (x)+g ′ (x)
(cf) ′ (x) =cf ′ (x)
(fg) ′ (x) =f ′ (x)g(x)+f(x)g ′ (x)
�

g(x) �= 0 x ∈ D f/g
� � ′
f
(x) =
g
f ′ (x)g(x) − f(x)g ′ (x)
g(x) 2
(f + g) ′ (f + g)(x + h) − (f + g)(x)
(x) := lim
h→0
h
f(x + h) − f(x) g(x + h) − g(x)
= lim
+lim
h→0 h
h→0 h
= f ′ (x)+g ′ (x)
(cf) ′ (cf)(x + h) − (cf)(x)
(x) := lim
n→∞
x
= c lim
n→∞
= cf ′ (x)
h
f(x + h) − f(x)
h
(fg) ′ =
(x)
(f · g)(x + h) − (f · g)(x)
lim
h→0
h
⎛
⎞
=
1
lim ⎝f(x + h)g(x + h) −f(x + h)g(x)+f(x + h)g(x) −f(x)g(x) ⎠
h→0 h
� �� �
=0
=
g(x + h) − g(x)
f(x + h) − f(x)
lim f(x + h) +limg(x)
h→0 h
h→0 h
= f(x)g ′ (x)+g(x)f ′ (x)

x
� � ′
f
g
1
= lim
h→0 h
(x)
�
f(x + h) f(x)
−
g(x + h) g(x)
1 f(x + h)g(x) − f(x)g(x + h)
= lim
h→0 h g(x + h)g(x)
�
� �� �
=
f(x + h)g(x) −f(x)g(x)+f(x)g(x) −f(x)g(x + h)
lim
h→0
hg(x)g(x + h)
= g(x) f(x + h) − f(x)
lim
g(x) 2 h→0 h
= f ′ (x)g(x) − f(x)g ′ (x)
g(x) 2
f : R → R,x↦→ cx
f : R → R,x↦→ x 2
f : R\{0} →R,x↦→ 1
x
fn : R → R,x↦→ x n
gn : R → R,x↦→ 1
x n
f : R → R,x↦→ c
=0
f ′ : R → R,x↦→ f ′ (x) =0
f ′ : R → R,x↦→ f ′ (x) =c
f ′ : R → R,x↦→ f ′ (x) =2x
f ′ : R → R,x↦→ f ′ (x) =− 1
x 2
f ′ n(x) =n · x n−1
g ′ n(x) =−n 1
x n+1
− f(x) g(x + h) − g(x)
lim
g(x) 2 h→0 h

f ′ f(x + h) − f(x) c − c
(x) := lim
= lim = lim
h→0 h
h→0 h h→0 0=0
f ′ f(x + h) − f(x) c(x + h) − cx
(x) := lim
= lim
= lim c = c
h→0 h
h→0 h
h→0
f ′ f(x + h) − f(x) (x + h)
(x) := lim
= lim
h→0 h
h→0
2 − x2 = lim (2x + h) =2x
h
h→0
f ′ � �
f(x + h) − f(x) 1 1 1
(x) := lim
= lim −
h→0 h
h→0 h x + h x
n =1 f ′ 1(x) =1
n → n +1
x − (x + h) −1 −1
= lim
= lim =
h→0 h(x + h)x h→0 (x + h)x x2 (fn+1) ′ (x) = (x n+1 ) ′ =(fn(x)f1(x)) ′
c)
= f ′ n(x)f1(x)+fn(x)f1(x) ′
g ′ n(x) d)
= nx n−1 · x + x n · 1
= (n +1)x n
= −nxn−1
(xn 1
= −n
) 2
x n+1
f :[a, b] → R f −1 :[A, B] →
[a, b] x ∈ (a, b) f ′ (x) �= 0 f −1
y = f(x)
(f −1 ) ′ (y) = 1
f ′ (x) =
1
f ′ (f −1 (y))
limn→∞ yn = y ∀n ∈ N : yn �= y
xn := f −1 (yn) x := f −1 (y)

f −1 xn �= x
f −1
g ◦ f
lim
n→∞ xn = lim
n→∞ f −1 (yn) =f −1 (y) =x
f
lim
n→∞
−1 (yn) − f −1 (y)
yn − y
=
xn − x
lim
n→∞ f(xn) − f(x)
= lim
n→∞
1
f ′ (x)�=0
=
1
f ′ (x)
f(xn)−f(x)
xn−x
f : D ⊂ R → R g : E ⊂ R → R f(D) ⊂ E
x ∈ D y = f(x) ∈ E
g ◦ f
(g ◦ f) ′ (x) =g ′ (f(x)) · f ′ (x)
limn→∞ xn = x xn �= x f(xn) �= f(x)
g(f(xn)) Def
= g(f(x)) + g ′ (f(x)) · (f(xn) − f(x)) + r1(f(xn))
Def
′ ′
= g(f(x)) + g (f(x)) · [f (x)(xn − x)+r2(xn)] + r1(f(xn))
= g(f(x)) + g ′ (f(x)) · f ′ (x)(xn − x)+r3(xn)
(cn)n
r3(xn) = g ′ (f(x))r2(xn)+r1(f(xn))
r1(f(xn)) = cn(f(xn) − f(x))
lim
n→∞ cn = 0

g ◦ f
r3(xn)
lim
n→∞ xn − x
g
= lim
n→∞
′ (f(x))r2(xn)+r1(f(xn))
xn − x
= g ′ (y) lim
n→∞
r2(xn)
xn − x
� �� �
=0
= lim
n→∞ cn
f(xn) − f(x)
xn − x
= f ′ (x) lim
n→∞ cn
= 0
+ lim
n→∞
r1(f(xn))
xn − x

⇐⇒
f :(a, b) → R x ∈ (a, b)
∃ε >0 ∀y ∈ B(x, ε) : f(x) ≥ f(y)
x ∈ (a, b) ⇐⇒
∃ε >0 ∀y ∈ B(x, ε) : f(x) ≤ f(y)
f :(a, b) → R x ∈ (a, b)
a) x ⇒ f ′ (x) =0
b) x �⇐ f ′ (x) =0
f ′ (x) =0
≤0
� �� �
f(y) − f(x)
lim
y→x,y>x y − x
� �� �
>0
≤0
� �� �
f(y) − f(x)
lim
y→x,y0
f(x) < 0 x

f(a)
(a, b)
f :[a, b] → R f(a) =f(b) (a, b)
y ∈ (a, b)
f ′
f ′ (y) =0
f ′ ≡ 0
∀y ∈ (a, b) :f ′ (y) =0
x0 ∈ (a, b) f(x0) >f(a) f(x0) <
f ′ (y) =0
c, d f :[a, b] → R f :[c, d] → R
f(c) =0=f(d)
y ∈ (c, d) f ′ (y) =0
f :[a, b] → R (a, b)
y ∈ (a, b)
f(b) − f(a)
= f
b − a
′ (y)
p ∈ (a, b)
F :[a, b] → R,x↦→ f(x) −
[a, b] (a, b)
y ∈ (a, b)
F (a) =f(a) =F (b)
0=F ′ (y) =f ′ (y) −
f :[a, b] → R (a, b)
a ≤ x ≤ y ≤ b
m ≤ f ′ (p) ≤ M
f(b) − f(a)
(x − a)
b − a
f(b) − f(a)
b − a
m(y − x) ≤ f(y) − f(x) ≤ M(y − x)
f
y ∈

⇐
x = y
x 0 ⇒
∀x ∈ (a, b) :f ′ (x) > 0 �⇐
y>x ⇒ m =0
f(y) − f(x) ≥ 0
f(y) − f(x)
y − x
� �� �
>0
f(x) ≤ f(y)
= f ′ (p) ≥ 0
f ′ ≥0
� �� �
f(y) − f(x)
(x) = lim
≥ 0
y↘x y − x
� �� �
>0

x>y≥ 0 x 3 >y 3
0 >x>y −y >−x >0
x>0 ≥ y x 3 > 0 ≥ y 3
x ≥ 0 >y x 3 ≥ 0 >y 3
f(y) − f(x)
= f
y − x
� �� �
>0
′ (p) > 0
f(y) >f(x)
f : R → R,x↦→ x 3
(−y) 3 > (−x) 3 > 0
−y 3 > −x 3 > 0
x 3 > y 3
f ′ : R → R,x↦→ 3x 2
f ′ (0) = 3 · 0 2 =0
f :(a, b) → R x ∈ (a, b)
(f ′ (x) =0 f ′′ (x) > 0) ⇒
(f ′ (x) =0 f ′′ (x) < 0) ⇒
f ′′ (x) > 0
f ′′ f
(x) = lim
y→x
′ (y) − f ′ (x)
> 0
y − x
∃ε >0 ∀y ∈ B(x, ε) : f ′ =0
� �� �
(y) − f ′ (x)
> 0
y − x
f ′ (y) < 0 x − ε

f ′′ (x) < 0
[x − ε, x] [x, x + ε]
f,g :[a, b] → R (a, b)
∀x ∈ (a, b) :g ′ (x) �= 0
g(b) �= g(a)
∃s ∈ (a, b)
g ′ (x) �= 0
g ′ (t) �= 0
g(a) �= g(b)
f(b) − f(a)
g(b) − g(a) = f ′ (s)
g ′ (s)
g(a) =g(b) t ∈ (a, b)
F (x) :=f(x) −
f ′ (s)
g ′ (s)
limx→a f ′ (x)
g ′ (x)
g ′ (t) =0
f(b) − f(a)
(g(x) − g(a))
g(b) − g(a)
F (b) = f(a) =F (a)
f(b) − f(a)
g(b) − g(a) g′ (x)
F ′ (x) = f ′ (x) −
s ∈ (a, b) F ′ (s) =0
0 = F ′ (s) =f ′ (s) −
= f(b) − f(a)
g(b) − g(a)
f(b) − f(a)
g(b) − g(a) g′ (s)
f,g :(a, b) → R ∀x ∈ (a, b) :
lim f(x) = 0 = lim
x→a x→a g(x)
limx→a f(x)
g(x)
f(x) f
lim = lim
x→a g(x) x→a
′ (x)
g ′ (x)

f(a) =g(a) =0
∀x ∈ (a, b) ∃s ∈ (a, x)
a ≤ s ≤ x
f(x)
g(x)
=0
����
f(x) − f(a)
=
g(x) − g(a)
����
=0
= f ′ (s)
g ′ (s)
f(x) f
lim = lim
x→a g(x) x→a
′ (s)
g ′ f
= lim
(s) s→a
′ (s)
g ′ (s)

u : [a, b] → R
⇐⇒ [a, b]
T [a, b]
a = x0

j�
i=1
j�
i=1
jk−1 ≤ i ≤ jk − 1
ei(ti − ti−1) ei=ck
=
jk−1 ≤ i ≤ jk − 1
ei(ti − ti−1) dk=ei
=
u ≤ v
a = t0

� b
a
� b
a
� b
a
(u + v)(x)dx =
(cu)(x)dx =
u(x)dx =
k=1
=
=
n�
(ak + bk)(xk − xk−1)
k=1
n�
ak(xk − xk−1)+
k=1
� b
a
u(x)dx +
� b
a
n�
bk(xk − xk−1)
k=1
v(x)dx
n�
n�
cak(xk − xk−1) =c ak(xk − xk−1) =c
u ≤ v ak ≤ bk
� b∗
a
� b
a∗
n�
k=1
ak≤bk
ak (xk − xk−1) ≤
� �� �
≥0
f :[a, b] → R
k=1
n�
k=1
bk (xk − xk−1) =
� �� �
≥0
� b
�� b
�
f(x)dx =inf u(x)dx : u ∈ T [a, b],u≥ f
a
�� b
�
f(x)dx =sup u(x)dx : u ∈ T [a, b],u≤ f
a
∀x ∈ [a, b] :−K ≤ f(x) ≤ K
a
� b
a
u(x)dx
v(x)dx

u = K1 [a,b]
� b∗
a
� b
a∗
f(x)dx ≤
f(x)dx ≥
u ∈ T [a, b]
� b∗
a
u = −K1 [a,b]
u(x)dx =
� b
a
� b
a
� b
a∗
Kdx = K(b − a) < ∞
Kdx = −K(b − a) > −∞
u(x)dx =
v ∈ T [a, b] v ≥ u
� b
a
� b
inf
v∈T [a,b],v≥u a
u ∈ T [a, b] u ≥ u
inf
v∈T [a,b],v≥u
� b
� b
v ∈ T [a, b] v ≤ u
a
� b
a
a
vdx ≥
u(x)dx =
v(x)dx ≤
� b
sup
v∈T [a,b],v≤u a
� b
a
v(x)dx ≥
udx
v(x)dx u≤u
≤
� b∗
a
� b
a
v(x)dx ≤
� b
� b
a
a
� b
a
u(x)dx
u(x)dx
� b
a
u(x)dx
u(x)dx
u(x)dx
v ≤ u
u(x)dx

u ∈ T [a, b] u ≤ u
�
sup
v∈T [a,b],v≤u
� b
a
� b
a∗
u(x)dx =
f(x)dx ≤
u(x)dx u≥u
≥
� b
a∗
� b∗
u, v ∈ T [a, b] u ≤ f ≤ v
� b
a
sup
u∈T [a,b],u≤f
sup
u∈T [a,b],u≤f
� b∗
a
c ≥ 0
� b
� b
a
a ∗
u(x)dx ≤
� b
f,g :[a, b] → R
a
a
� b
a
u(x)dx ≤
�
u(x)dx
f(x)dx
v(x)dx
� b
u(x)dx ≤ inf
v∈T [a,b],v≥f
f(x)dx ≤
(f + g)(x)dx ≤
� b∗
a
� b∗
a
(cf)(x)dx = c
� b ∗
a
a
f(x)dx
f(x)dx +
� b∗
a
u(x)dx
v(x)dx
� b
a
� b∗
a
f(x)dx
v(x)dx
g(x)dx

u, v ∈ T [a, b] u ≥ f v ≥ g
� b
a
� b
a
u(x)dx ≤
v(x)dx ≤
u + v ≥ f + g ε>0
ε>0
� b∗
a
(f + g)(x)dx inf
≤
� b∗
a
� b∗
a
=
≤
� b
a
� b
(f + g)(x)dx ≤
a
� b∗
a
cf ≡ 0 ∈ T [a, b]
(cf)(x)dx =
u ∈ T [a, b] u ≥ f
c>0 cu ≥ cf
� b∗
a
� b
a
� b∗
a
u(x)dx ≤
� b∗
a
� b∗
a
(u + v)(x)dx
u(x)dx +
f(x)dx + ε
2
g(x)dx + ε
2
� b
a
f(x)dx + ε
2 +
� b∗
a
f(x)dx +
0dx =0= c
����
=0
� b
� b∗
a
v(x)dx
� b∗
a
� b∗
a
� b∗
a
f(x)dx + ε
c
� b
g(x)dx + ε
2
g(x)dx
f(x)dx
(cf)(x)dx inf
≤ (cu)(x)dx = c u(x)dx
≤
a
a
�� b∗
c f(x)dx + ε
�
c
= c
a
� b∗
a
f(x)dx + ε

ε>0
cv ≥ cf
ε>0
� b∗
� b∗
c>0 v ≥ f
c
� b∗
a
a
a
(cf)(x)dx ≤ c
(cv)(x)dx ≤
f(x)dx inf
≤ c
c
� b∗
a
≤
� b
a
� b∗
a
f(x)dx ≤
� b∗
a
� b∗
a
f(x)dx
(cf)(x)dx + ε
v(x)dx =
� b
a
(cf)(x)dx + ε
� b∗
a
(cf)(x)dx
(cv)(x)dx
− sup{−f(x) :x ∈ A} =inf{f(x) :x ∈ A}
K =inff(x)
x∈A
⇐⇒ ∀x ∈ A : f(x) ≥ K
∀L >K ∃x ∈ A : L>f(x)
⇐⇒ ∀x ∈ A : −f(x) ≤−K
∀−L −L
⇐⇒ −K =sup−f(x)
x∈A
f :[a, b] → R
−
� b
a∗
(−f)(x)dx =
� b∗
a
f(x)dx
v ∈ T [a, b]

� b
a∗
� b∗
a
c

� b∗
a
� b
a∗
cf(x)dx =
cf(x)dx =
⇐⇒
� b∗
a
� b
a∗
(−c)(−f)(x)dx −c≥0
= −c
(−c)(−f)(x)dx −c≥0
= −c
� b∗
f :[a, b] → R
⇐⇒
∀ε >0 ∃u, v ∈ T [a, b] :u ≤ f ≤ v
⇒
v ∈ T [a, b] v ≥ f
u ≤ f
≤
a
� b
a
� b
a
� b
a
= ε
� b
a
� b∗
a
f(x)dx =
v(x)dx −
v(x)dx ≤
u(x)dx ≥
v(x)dx −
� b
a
� b∗
a
� b
a∗
� b
a
� b
a∗
f(x)dx + ε
2 −
� b∗
a
� b
a∗
f(x)dx
u(x)dx ≤ ε
f(x)dx + ε
2
f(x)dx − ε
2
u(x)dx
� b
a∗
� b
(−f)(x)dx = c f(x)dx
a∗
(−f)(x)dx = c
f :[a, b] → R
f(x)dx + ε
2
� b∗
a
f(x)dx
u ∈ T [a, b]

⇐
ε>0
0 ≤
� b∗
a
= inf
v≥f
≤
≤ ε
� b
a
� b∗
a
� b
f(x)dx −
� b
a
a∗
v(x)dx − sup
v(x)dx −
f(x)dx =
� b
a
� b
a∗
f(x)dx
u≤f
� b
a
u(x)dx
f(x)dx
f :[a, b] → R
u(x)dx
[a, b]
[a, b] ∀ε >0 ∃δ >0 ∀x, y
x, y ∈ (ck,ck+1)
|x − y|

=
= b − a
= b − a
= b − a
u, v ∈ T [a, b] x ∈ [a, b]
u(x) ≤ f(x) ≤ v(x)
v(x) − u(x) ≤
ε
b − a
=
≤
� b
a
� b
a
� b
a
v(x)dx −
� b
a
u(x)dx
( v(x) − u(x) )dx
ε
= ε
b − a
b − a
xk = a + k
n
u, v ∈ T [a, b]
u(x) := f(xk−1) xk−1 ≤ x

f + g cf
c

g − f
g − f ≥ 0 0 ∈ T [a, b] 0 ≤ g − f
f+
|f|
=
� b
a
� b
a
= sup
u≤g−f
g(x)dx −
� b
a
(g − f)(x)dx =
� b
a
� b
a
f(x)dx
� b
a∗
u(x)dx 0≤g−f
≥
f(x)dx ≤
� b
a
(g − f)(x)dx
� b
a
g(x)dx
0dx =0
f : D ⊂ R → R
�
f(x)
f+ : D ⊂ R → R, x ↦→
0
f(x) > 0
f(x) ≤ 0
�
−f(x)
f− : D ⊂ R → R, x ↦→
0
f(x) < 0
f(x) ≥ 0
f = f+ − f−
|f| = f+ + f−
f+(x) − f−(x) =
�
f(x)+0
0 − (−f(x))
f(x) > 0
f(x) ≤ 0
f+(x)+f−(x)
=
=
f(x)
�
f(x)+0
0+(−f(x))
f(x) > 0
f(x) ≤ 0
= |f(x)|
f−
f,g :[a, b] → R
p ∈ N |f| p
f · g :[a, b] → R

u−,v−
v(x)
u+,v+
f+
� b
a
� b
a
ε>0 u, v ∈ T [a, b] u ≤ f ≤ v
� b
a
(v+ − u+)(x)dx ≤
(v− − u−)(x)dx ≤
(v − u)(x)dx ≤ ε
� b
a
u+ ≤ f+ ≤ v+
(v − u)(x)dx ≤ ε
u− ≤ f− ≤ v−
� b
a
(v − u)(x)dx ≤ ε
f−
|f| = f+ + f−
0 ≤ f ≤ 1
ε>0 u, v ∈ T [a, b] 0 ≤ u ≤ f ≤ v ≤ 1
x ∈ [a, b]
u p v p
� b
a
s c d
d p − c p
d − c
(v − u)(x)dx ≤ ε
p
0 ≤ u p ≤ f p ≤ v p ≤ 1
g : R → R,y ↦→ y p
d
g(y) =pyp−1
dy
= g(d) − g(c)
d − c
= g ′ (s) =ps p−1
c = u(x) d = v(x) s u(x)
u(x) p − v(x) p
u(x) − v(x)
= psp−1

|f| p
u(x),v(x) ∈ [0, 1] s ∈ [0, 1]
v(x) p − u(x) p
� b
a
(v p − u p )(x)dx ≤ p
f |f|
f K ∈ R
|f|
K
p =2
= p(v(x) − u(x))s p−1
0≤s≤1
≤ p(v(x) − u(x))
0 ≤ |f|
K
� b
a
≤ 1
|f| = |f|p
Kp
Kp (v − u)(x)dx ≤ ε
fg = 1 � 2 2
(f + g) − (f − g)
4
�
f :[a, b] → R g :[a, b] → [0, ∞)
∃p ∈ [a, b] :
� b
a
∃p ∈ [a, b] :
f(x)g(x)dx = f(p)
� b
a
� b
f(x)dx = f(p)(b − a)
f · g
m := inf
x∈[a,b] f(x)
M := sup
x∈[a,b]
f(x)
mg ≤ fg ≤ Mg
a
g(x)dx

m
1 ∈ T [a, b]
� b
a
g(x)dx =
≤
s ∈ [m, M]
� b
a
� b
a
� b
a
� b
a
mg(x)dx ≤
f(x)g(x)dx = s
� b
a
Mg(x)dx = M
� b
a
f(x)g(x)dx
� b
g(x)dx
p ∈ [a, b] f(p) =s
1dx =1(b − a) =b − a
f :[a, b] → R
[a, b]
a = x0 0 x0,...,xn
s ≤ δ
�
��
�
b
n�
�
�
�
� f(x)dx − f(yk)(xk − xk−1) � ≤ ε
�
�
a
yk ∈ [xk−1,xk]
k=1
K := sup
x∈[a,b]
|f(x)| < ∞

ε>0 u, v ∈ T [a, b] u ≤ f ≤ v
� b
a
(v − u)(x)dx ≤ ε
2
u, v a = t0

� b
a
2m r(x) �= 0
� b
a
(u(x) − r(x))dx ≤
� b
a
ti ∈ (a, b)
u(x)dx − ε
2 ≤
−ε ≤
−ε ≤
≤
≤
≤
≤
≤ ε
� b
a
� b
r(x)dx ≤ 2K2ms s≤δ
≤ ε
2
� b
a
w(x)dx ≤
� b
n�
f(yk)(xk − xk−1) ≤
k=1
� b
a
(v(x)+r(x))dx
� b
u(x)dx − v(x)dx −
a
ε
2
� b
u(x)dx − f(x)dx −
a
a
ε
2
n�
� b
f(yk)(xk − xk−1) − f(x)dx
k=1
� b
a
� b
a
v(x)dx −
v(x)dx −
� b
a
� b
n�
f(yk)(xk − xk−1) −
k=1
1ti
a
a
f(x)dx + ε
2
u(x)dx + ε
2
� b
�
1 x = ti
:[a, b] → R,x↦→
0
� b
a
1ti (x)dx =0
a
a
f(x)dx ≤ ε
v(x)dx + ε
2

� b
a
a = x0

� c
a
ui,vi
(v − u)dx =
u :=
v :=
�
u1 + u2 x �= b
min(u1,u2)
�
v1 + v2
x = b
x �= b
max(v1,v2) x = b
u ≤ f ≤ v
m�
(ei − di)(ti − ti−1)+
i=1
� b
= (v1 − u1)dx +
a
< ε ε
+ = ε
2 2
� c
b
m+n �
i=m+1
(v2 − u2)dx
⇒ tm
[a, c] u, v ∈ T [a, c]
� b
a
� c
b
(v − u)(x)dx =
u ≤ f ≤ v
�
(v − u)dx < ε
≤
(v − u)(x)dx =
≤
m�
(ei − di)(ti − ti−1)
i=1
(ei − di)(ti − ti−1)
m+n �
(ei − di)(ti − ti−1)

N
−ε <
� b
a
=
≤
≤
=
� c
a
� b
a
� b
a
� b
a
� c
a
fdx+
(u − v)dx
udx +
fdx+
vdx +
� c
b
� c
b
� c
b
udx −
fdx−
vdx −
(v − u)dx < ε
� c
b
fdx =
� c
a
fdx
� c
a
� c
a
� c
a
vdx
fdx
udx
f :[a, ∞) → R [a, N] N ∈
� ∞
f(x)dx = lim
a
� a
−∞
� ∞
−∞
� b
a
� b
a
� N
N→∞ a
� a
f(x)dx = lim
N→∞
f(x)dx = lim
N→∞
� b
f(x)dx = lim
ε→0
f(x)dx = lim
ε→0
−N
� N
−N
a+ε
� b−ε
a
f(x)dx
f(x)dx
f(x)dx
f(x)dx
f(x)dx

I ⊂ R
f : I → R a ∈ I x ∈ I
F : I → R,x↦→
F ′ = f
� x
h �= 0 x, x + h ∈ I
yh ∈ [x, x + h]
F (x + h) − F (x)
h
=
=
1
h
1
h
a
f(t)dt
�� x+h
f(t)dt −
a
� x+h
x
f(t)dt
� x+h
∃yh∈[x,x+h]
=
1
h f(yh)
x
1dt
=
1
h f(yh)(x + h − x)
= f(yh)
� x
F (x + h) − F (x)
lim
= lim f(yh) = f(x)
h→0 h
h→0
f : I → R
F ′ = f
a
F : I → R
F : I → R f : I → R
f ⇐⇒ F − G
⇒ G ′ = f = F ′
F − G = c
⇒ F − G = c
(F − G) ′ = F ′ − G ′ =0
G ′ =(F − c) ′ = F ′ = f
�
f(t)dt

f : I → R
a, b ∈ I
F (b) − F (a) =
F0 : I → R,x↦→
� b
a
� x
a
f(t)dt
f(t)dt
F0 F0(a) =0 F0(b) = � b
a f(t)dt
c ∈ R F − F0 = c
F (b) − F (a) = F0(b)+c − (F0(a) +c)
� �� �
=0
= F0(b) =
� b
a
f(t)dt
f : I → R g :[a, b] → I ⊂ R
f ◦ g :[a, b] → R
� b
a
[a, b]
f◦g
−→ R
g ↘ ↗ f
I
f(g(t))g ′ (t)dt =
� g(b)
g(a)
f(x)dx
f,g,g ′ (f ◦ g)g ′ f
F : I → R,a↦→
� x
a
f(t)dt
(F ◦ g) ′ (t) =F ′ (g(t)) · g ′ (t) =f(g(t)) · g ′ (t)
F ◦ g :[a, b] → R

� b
a
� b
a
f(g(t)) · g ′ (t)
f(g(t))g ′ (t)dt =
f,g :[a, b] → R
� b
a
(F ◦ g) ′ (t)dt
= F (g(b)) − F (g(a))
=
� g(b)
g(a)
f(x)dx
f(x) · g ′ (x)dx = f(b)g(b) − f(a)g(a) −
� b
(fg) ′ (x) =f ′ (x)g(x)+f(x)g ′ (x)
fg f ′ (x)g(x)+f(x)g ′ (x)
=
=
� b
f ′ (x)g(x)dx +
� b
a
a
� b
(f
a
′ (x)g(x)+f(x)g ′ (x)) dx
� b
a
(f · g) ′ (x)dx
= f(b)g(b) − f(a)g(a)
a
f(x)g ′ (x)dx
g(x)f ′ (x)dx

⇐⇒
⇐⇒
R
(a0,a1,...) R
�
n�
(a0,a0 + a1,a0 + a1 + a2,...)=
k=0
� ∞
k=0 ak � ∞
k=0 ak
� ∞
k=0 ak
(ak)k
�
� n� �
∀ε >0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : �
�
� n�
k=0
� n�
k=0
ak
ak
�
�
n
n
k=m
�
� n� �
⇐⇒ ∀ε>0 ∃n0 ∈ N ∀n, m ≥ n0 : �
�
k=0
�
� n� �
⇐⇒ ∀ε>0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : �
�
� ∞
n=0 an
m = n − 1
�
� n�
�
∀ε >0 ∃n0 ∈ N ∀n ≥ n0 : |an| = �
�
lim
n→∞ an =0
k=0
ak
ak
ak −
k=m
�
n
�
�
�
�
�

⇒
⇐
�
�n+1
��
�
�
k=1
ak −
� ∞
n=0 an
( � n
k=0 ak)n
n�
k=1
ak
∞�
n=0
(−1) n
( � n
k=1 ak) n
�
�
�
�
� = |an+1| = |(−1) n+1 | =1
an ≥ 0
⇐⇒
an ≥ 0 ( � n
k=0 ak)n
�∞ k=0 |ak|
�∞ n=0 |an|
� ∞
k=0 ak.
� �
� n� �
� �
∀ε >0 ∃n0 ∀n ≥ m ≥ n0 : � |ak| �
� � 0 ∃n0 ∈ N ∀n, m ≥ n0
�
� n�
�
�
� �
� ak�
� � ≤
n�
�
� n�
�
�
� �
|ak| = � |ak| �
� �

n0
� ∞
n=0 an
S : N → N �∞
n=0 |an|
�
�
�
�a
−
�
∃n0 :
n0−1 �
k=0
ak
∞�
k=n0
|an|
�
�
�
�
� ≤
|ak| < ε
2
∞�
k=n0
aj = a S(S −1 (j))
|ak| < ε
2
N := max
1≤j≤n0−1 S−1 (j)
{a1,...,an0−1} = {a S(S −1 (1)),...,a S(S −1 (n0−1))}
�
� m� �
�a
−
�
k=0
m ≥ N
�
�
�
�
� ≤
a S(k)
an
� ∞
n=0 cn
⊂ {a S(1),...,a S(N)}
�
�
�
�a
−
�
≤ ε
2 +
n0−1 �
k=0
ak
�
�
�
�
� +
�
�
� �
�
�
∞�
|ak| 0 ∃n0 ∀n ≥ m ≥ n0 : �
�
k=m
ck
�
�
�
�
�

n → n +1
∀ε >0 ∃n0 ∀n ≥ m ≥ n0
�
� n�
�
�
� �
� ak�
� � ≤
n�
|ak| 0≤|an|≤ck
�
� n� �
≤ �
�
n =1
k=m
k=m
x �= 1 n ∈ N
�
n+1
x k
k=0
|x| < 1
1
|x| > 1
1+x =
n�
x k =
k=0
1 − xn+1
1 − x
(1 + x)(1 − x)
1 − x
= x n+1 +
n�
k=0
= 1 − x2
x k
k=m
ck
1 − x
�
�
�
�
� 0
|x|
1+y =
1
> 1
|x|
1
|x| n = (1+y)n ≥ 1+ny
> ny > 1
ε

lim
n→∞
k=0
j =1
j → j +1
∀ε >0 ∃n0 ∈ N ∀n ≥ n0 : |x n − 0| = |x| n

(s2k)k
m, n ≥ m0
(s2k+1)k
n�
|ak| =
k=m
(an)n
≤
n�
k=m
n�
k=m
|an0+k−n0 |
|an0 |qk−n0
≤ |an0 |qm−n0
≤ |an0 |qm−n0
= |an0 |
� ∞
k=0 (−1)k ak
0 ≤ a0 − a1 ≤
sk = � k
n=0 (−1)n an.
s2k+2 − s2k =
s2k+3 − s2k+1 =
s2k+1 ≤ s2k
s2k+1 − s2k =
2k+2
q n0
n−m �
i=0
∞�
i=0
q i
q i
1
1 − q qm

(s2k)k
(s2k)k
(s2k+1)k
(s2k+1)k
(s2k)k
s = s ′
(s2k+1)k
k ≥ max{n1,n2}
� ∞
n=0 |cn|
s2k ≥ s2k+1 ≥ s1
s2k+1 ≤ s2k ≤ s2
s − s ′ = lim
k→∞ s2k − lim
k→∞ s2k+1
= lim
k→∞ (s2k − s2k+1)
s
= lim
k→∞ a2k+1
Vor
= 0
∃n1 ∀k ≥ n1 : |s2k − s| < ε
∃n2 ∀k ≥ n2 : |s2k+1 − s| < ε
|sk − s|

∞�
cn =
� ∞�
n=0 n=0
�∞ n=0 |an|, �∞ n=0 |bn|
�∞ n=0 bn
Dn :=
|D|n :=
Qn :=
|Q|n :=
n�
k=0
an
ck =
n�
|ck|
k=0
� n�
k=0
ak
�� ∞�
n�
n=0
k�
k=0 j=0
�� n�
k=0
bn
�
ak−jbj
bk
�
�
n�
��
n�
�
|ak| |bk|
k=0
k=0
lim
n→∞ |Q|n
�
n�
��
n�
�
= lim |ak| |bk|
n→∞
k=0 k=0
�
n�
� �
n�
�
= lim |ak| lim |bk|
n→∞
n→∞
k=0
k=0
�
∞�
��
∞�
�
= |ak| |bk| < ∞
(|Q|n)n
∀ε >0 ∃n0 ∀n ≥ n0
k=0
k=0
||Q|n −|Q|n0 | = �
i,j≤n
|ai|·|bj|− �
i,j≤n0
|ai|·|bj|
< ε
2
� ∞
n=0 an

n>2n0
n>2n0
||Q|n −|D|n| = �
=
≤
0≤i,j≤n
�
0≤i,j≤n,i+j>n
�
n0≤i,j≤n
|ai| |bj|− �
|ai|·|bj|
i+j≤n
|ai|·|bj|
≤ ||Q|n −|Q|n0| < ε
2
|ai||bj|
lim
n→∞ |D|n = lim
n→∞ |Q|n < ∞
∞�
�
∞�
��
∞�
�
|cn| = |ak| |bk| < ∞
n=0
|Qn − Dn| =
=
=
≤
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
k=0
�
0≤i,j≤n
�
0≤i,j≤n
�
aibj −
k=0
n�
k�
k=0 j=0
aibj − �
0≤i,j≤n,i+j>n
�
n0≤i,j≤n
i+j≤n
aibj
|ai|·|bj|
ε
= ||Qn|−|Qn0 || <
2
lim
n→∞ Qn = lim
n→∞ Dn
�
�
�
�
�
�
ak−jbj
aibj
�
�
�
�
�
�
�
�
�
�
�
�

∞�
k=0
ck
Def
= lim
n→∞ Dn
= lim
n→∞ Qn
�
n�
= lim
n→∞
k=0
�
n�
= lim
n→∞
k=0
�
∞�
=
n=0
an
ak
ak
�� n�
�
�� ∞�
n=0
k=0
lim
n→∞
bn
�
bk
�
� n�
k=0
bk
�

� � n
k
n! := 1· 2 · ...· n
�
0!
�
n
k
:=
:=
1
n!
(n − k)!k!
� �
n
n
� �
n
0
� �
n +1
k
� �
n
n
� �
n
0
� � � �
n n
+
k k − 1
n�
k=0
=
=
=
= n!(n − k +1)+n!k
=
= 1
= 1
=
� n
k
�
+
�
n
k − 1
�
n! 1
=
(n − n)!n! 0! =1
n!
n!(n − n)! =1
n!
k!(n − k)! +
n!
(k − 1)!(n − k +1)!
k!(n − k +1)!
(n +1)!
k!(n +1− k)! =
� �
n +1
k
� �
n
x
k
n−k y k =(x + y) n

n → n +1
(x + y) n+1
n =1
x + y = x 1−0 · y 0
����
=1
+ x 0
����
=1
·y 1−0
= (x + y) n (x + y)
�
n� � �
n
=
x
k
k=0
n−k y k
�
(x + y)
n�
� �
n
=
x
k
k=0
n+1−k y k n�
� �
n
+ x
k
k=0
n−k y k+1
= x n+1 n�
� �
n
+ x
k
k=1
n+1−k y k n−1 �
� �
n
+ x
k
k=0
n−k y k+1 + y n+1
= x n+1 n�
� �
n
+ x
k
k=1
n+1−k y k n�
� �
n
+
x
k − 1
k=1
n−(k−1) y k + y n+1
� �
n +1
=
x
0
n+1 n�
� �
n +1
+
x
k
k=1
n+1−k y k � �
n +1
+ y
n +1
n+1
n+1 �
� �
n +1
=
x
k
n+1−k y k
k=0

n ≥ 2|x|
x ∈ R
e x := exp(x)
�
�
�
�
an+1
an
�∞ |x|
n=0
n
n! �∞
n=0 xn
n!
∀|x| ≤
exp(x)
R
exp : R → R,x↦→
e := exp(1) =
�
�
�
� = |x|n+1n! |x| n (n +1)!
exp(x) =
N +2
2
N�
n=0
x n
n!
�∞ |x|
n=0
n
n!
∞�
n=0
∞�
n=0
1
n!
x n
n!
|x| 1
= ≤
n +1 2
+ rN+1(x)
: |rN+1(x)| ≤2 |x|N+1
(N +1)!

|rN+1(x)| =
�∞ |x|
n=0
n
n!
=
N+2≤N+k+1
≤
|x| 1
N+2 ≤ 2
≤
=
x, y ∈ R
∞� |x|
n=N+1
n
n!
|x| N+1
�
∞� |x|
1+
(N +1)!
k=1
k
�
(N +2)···(N + k +1)
|x| N+1 ∞� |x|
(N +1)!
k
(N +2) k
|x| N+1
(N +1)!
|x| N+1
(N +1)!
k=0
∞�
k=0
1
1
2 k
1 − 1
2
=2 |x|N+1
(N +1)!
exp(x)exp(y) =exp(x + y)
cn :=
�∞ |y|
n=0
n
n!
exp(x)exp(y) =
n� x
k=0
n−kyk (n − k)!k!
= 1
n�
� �
n
x
n! k
n−k y k
k=0
= 1
(x + y)n
n!
=
∞�
n=0
∞�
n=0
cn
1
n! (x + y)n =exp(x + y)

n ∈ Z
x ∈ R
exp(0) = 1
exp(−x) =
1
exp(x)
exp(x) > 0
exp(n) =e n
exp(0) =
∞� 0
1 +
n=1
n
n! =1
exp(x)exp(−x) = exp(x−x) =exp(0)
exp(x) �= 0 exp(x)
x ≥ 0
x 0
1
> 0
exp(−x)
exp(n +1)=exp(n)exp(1)=e n · e = e n+1
n ∈ N
exp(−n) =
exp : R → R,x↦→ exp(x)
exp 0
limn→∞ xn =0
1
exp(n)
exp(x) = 1+r1(x)
1
= = e−n
en |r1(x)| ≤ 2|x| |x| < 0+2
2

exp
0 ≤ lim
n→∞ | exp(xn) − 1|
|xn|

exp
x ∈ R
exp ′ exp(x + h) − exp(x)
(x) := lim
h→0
h
exp(x)exp(h) − exp(x)
:= lim
h→0
h
exp(h) − 1
= exp(x) lim
=exp(x)
h→0 h
c ∈ R f : R → R
∀x ∈ R : f ′ (x) = cf(x)
a := f(0)
∀x ∈ R : f(x) =ae cx
F : R → R,x↦→ f(x)e −cx
F ′ (x) = f ′ (x)e −cx − cf(x)e −cx
= (f ′ (x) − cf(x))e −cx
= 0
F (0) = f(0) e −c0
����
=1
=1
F ≡ a
f(x) = ae cx
exp : R → (0, ∞)
f : R → (0, ∞) f ′ = f f(0) = 1
⇒ exp ′ (x) =exp(x) exp(0) = 1
⇐ c =1 A =1 f(x) =e x

R (0, ∞)
x, y ∈ (0, ∞)
exp : R → (0, ∞)
lim exp(n) = ∞
n→∞
lim
n→−∞
exp(n) = 0
ln : (0, ∞) → R
ln(xy)
∀k ∈ N : ln
= lnx +lny
� x k� = k ln x
ln(1)
� �
1
ln
x
=
=
0
− ln x
n ∈ N
x0
exp(n) ≥ 1+n
0 < exp(−n) =
1 1
≤
exp(n) 1+n
lim exp(n) = ∞
n→∞
lim
n→∞
exp(−n) = 0
∞� (y − x)
exp(y − x) =1+(y− x)+
k
> 1
k!
k=2
exp(y) − exp(x) = exp(y−x + x) − exp(x)
= exp(x)exp(y−x) − exp(x)
= exp(x) (exp(y − x) −1) > 0
� �� � � �� �
>0 >1

k =2
� �
1
, 1+n ⊂ [exp(−n), exp(n)]
1+n
�
[exp(−n), exp(n)] = (0, ∞)
n∈N
exp R (0, ∞)
ln : (0, ∞) → R
k → k +1
ln
exp(ln x +lny) =exp(lnx)exp(lny) =xy
ln
ln x +lny =lnxy
ln(x · x) =lnx +lnx
ln(x k+1 ) = ln � x · x k� =lnx k +lnx
= k ln x +lnx =(k +1)lnx
exp(0) = 1
ln(1) = 0
0 =
�
ln(1) = ln x · 1
�
x
= lnx +ln 1
x
ln 1
x
= − ln x
k ∈ N
limx→∞ ln x = ∞
limx↘0 ln x = −∞
∀k ∈ N :limx→0 x k ln x =0
exp(x)
lim
x→∞ xk = ∞
[−n, n] [exp(−n), exp(n)]

N ∈ N x>0
x>(k +1)!N
exp(x) ≥ xk+1
(k +1)!
exp(x)
x k
x>exp(N) ln x>N
≥
x
(k +1)! >N
lim ln x = ∞
x→∞
lim ln x
x↘0
=
1
lim ln
y→∞ y
= − lim ln y = −∞
y→∞
exp ln x k
x k = exp ln x k =exp(k ln x)
lim
x↘0 xk ln x = lim exp(k ln x)lnx
x↘0
= lim
y→−∞ exp(ky)y
= −1
k lim
y→∞ exp(−y)y
= −1
k lim
y
y→∞ exp(y)
a)
= 0
ln ′ (x) = 1
x
limn→∞ an = a ∃n0 ∀n ≥ n0 : an �= 0
�
lim 1+
n→∞
an
�n �
=exp
n
lim
n→∞ an
�

lim
n→∞
n ≥ n0
�
1+ an
n
� n
ln ′ (x) =
limn→∞ an = a �= 0
∀n ≥ n0
limn→∞ an = a �= 0
�
lim 1+
n→∞
1
�n = e
n
1
exp ′ (ln x) =
1 1
=
exp(ln x) x
an
= lim
n→∞ exp
� �
n ln 1+ an
��
n
⎛
⎜
= exp ⎜
⎝ lim
n→∞ an
�
ln 1+ an
⎞
=0
� � �� �
− ln(1) ⎟
n
⎟
⎠
exp stetig
⎛
= exp⎝lim
n→∞ an
�
ln 1+
lim
n→∞
an
� ⎞
− ln(1)
n ⎠
an
⎛
n
= exp⎝lim
n→∞ an · ln ′ ⎞
�
(1) ⎠ =exp lim
� �� �
n→∞
=1
an
�
∃n0 ∀n ≥ n0 : |an − a| < |a|
2
|an| = |a − a + an|
≥ |a|−|a− an|
> |a|− |a| |a|
=
2 2
∀n ∈ N : an =1
> 0
�
lim 1+
n→∞
1
�n =exp(1)=e
n
an
n

f :[1, ∞) → R+
∞�
� ∞
f(n) < ∞ ⇐⇒ f(x)dx < ∞
n=1
f(n) ≤ f(x) ≤ f(n − 1) x ∈ [n − 1,n]
f(n) =
≤
� n
n−1
� n
n−1
= f(n − 1)
N�
f(n) ≤
n=2
N�
f(n) ≤
n=2
n=2
f(n)dx ≤
1
� n
n−1
f(x)dx
f(n − 1)dx =(n − (n − 1))f(n − 1)
N�
� n
n=2 n−1
� N
1
1
f(x)dx ≤
f(x)dx ≤
N−1 �
n=1
N�
f(n − 1)
n=2
f(n)
∞�
� ∞
∞�
f(n) ≤ f(x)dx ≤ f(n)
� N
lim
N→∞ 1
∞�
n=1
lim
x→∞
1
= ∞
n
ln ′ (x) = 1
x
ln x = ∞
ln(1) = 0
1
dx = lim
x
N→∞
n=1
⎛ ⎞
⎝ln N − ln 1 ⎠
= lim ln N = ∞
N→∞
� ∞
1
dx = ∞
x
1
����
=0

� ∞
1
dx < ∞ ⇐⇒
x
1
∞�
n=1
1
< ∞
n

f : I ⊂ R → R
a, x ∈ I
f(x) = f(a)+ f ′ (a)
(x − a)+
1!
f ′′ (a)
2!
+Rn+1(x)
=
n�
n − 1 → n
Rn(x) =
k=0
n =0
f (k) (a)
(x − a)
k!
k + Rn+1(x)
� x
a
Rn+1(x) =
� x
a
f(x) = f(a)+
� � n ′
(x − t)
n!
1
(n − 1)!
= −f (n) (x)
= f(a)+
(x − a) 2 + ...+ f (n) (a)
(x − a)
n!
n
(x − t) n
f
n!
(n+1) (t)dt
� x
a
� x
a
f ′ (t)dt
(x − t) 0
f
� ��
0!
�
=1
(1) (t)dt
g ′ hdt = g(x)h(x) − g(a)h(a) −
= −(x − t)n−1
� �� �
=0
= f (n) (x − a)n
(a) +
n!
(n − 1)!
� x
� x
(x − t)
a
n−1 f (n) (t)dt
(x − x)n
+f
n!
(n) � x
(x − a)n
(a) +
n! a
� x
a
(x − t) n
f
n!
(n+1) (t)dt
a
gh ′ dt
(x − t) n
f
n!
(n+1) (t)dt
f : I ⊂ R → R a, x ∈ I
f(x) =
n�
k=0
f (k) (a)
k!
(x − a) k + f (n+1) (s)
(x − a)n+1
(n +1)!

f (n+1)
∀t ∈ [a, x] : (x − t) n ≥ 0
� x
Rn+1(x) = 1
(x − t)
n! a
n f (n+1) (t)dt
= f (n+1) (s) 1
� x
(x − t)
n! a
n dt
= −f (n+1) � �
n+1 (x − x) (x − a)n+1
(s)
−
(n +1)! (n +1)!
= f (n+1) (s)
(x − a)n+1
(n +1)!
f : I ⊂ R → R a ∈ I
x ∈ I
f(x) =
n+1 �
k=0
r(x)(x − a) n+1
f (k) (a)
(x − a)
k!
k + r(x)(x − a) n+1
lim r(x) =0
x→a
n+1 �
:= f(x) −
k=0
f (k) (a)
(x − a)
k!
k
= f (n+1) (s) − f (n+1) (a)
(x − a)
(n +1)!
n+1
limx→a
lims→a
lim
x→a r(x) =lims→a(f (n+1) (s) − f (n+1) (a))
=0
(n +1)!

K K = Q K = R
K
u, v, w ∈ V c, d ∈ K
(u + v)+w = u +(v + w)
v + w = w + v
∃0 ∈ V ∀v ∈ V : v +0=v
∀v ∈ V ∃−v ∈ V : v +(−v) =0
1 · v = v
(cd) · v = c · (d · v)
c(v + w) =cv + cw
(c + d)v = cv + dv
v + w, cw, c + d, c · d
+:V × V → V,(v, w) ↦→ v + w
· : K × V → V,(c, v) ↦→ cv
K n := (a1,...,an) ai ∈ K 1 ≤ i ≤ n
(a1,...,an)+(b1,...,bn) = (a1 + b1,...,an + bn)
c(a1,...,an) = (ca1,...,can)
(an)n +(bn)n = (an + bn)n
f : R → R
c(an)n = (can)n
(f + g)(x) := f(x)+g(x)
(cf)(x) = c · f(x)

c ∈ K
K K
(0,...,0) ∈ K n
{0}
0+0 = 0
c · 0 = 0
(0)n
(an)n (bn)n (an + bn)n (can)n
(0)n
(0)n
limn→∞ an =0=limn→∞ bn
0
����
∈K
lim
n→∞ (can) = c lim
n→∞ an =0
lim
n→∞ (an + bn) = lim
n→∞ an + lim
n→∞ bn =0+0=0
0:D → R,x↦→ 0
f,g : R → R f+g cf
f,g : R → R f + g cf
f,g : R → R f + g cf
f,g : R → R f + g cf
·v = 0
����
∈V
(−1) · v = −v
v ∈ V −v
0=0

+, ·
0 ∈ V
0 ∈ U
0 ′
0
0
= 0+0 ′ 0 ′
= 0 ′
w ∈ V w + v =0
w = w +0=w + v +(−v)
= 0+(−v) =−v
����
0 ·v =(
����
0 +
����
0 ) · v =
����
0 ·v +
����
0 ·v
∈K ∈K ∈K
∈K ∈K
0
����
∈V
0
����
∈K
·v = 0
����
∈V
c)
=
����
0 ·v =(1+(−1))v
∈K
= 1· v +(−1)v = v +(−1)v
−v −v =(−1) · v
⇐⇒
u1,u2 ∈ U u1 + u2 ∈ U
u ∈ U, c ∈ K c · u ∈ U
v ∈ V
K U ⊂ V
K · v = {c · v | c ∈ K}
{0}

U = {0}
Kv ⊂ V a, b, c ∈ K
0 ∈ U
0+0=0 ∈ U
∀c ∈ K : c · 0=0 ∈ U
0 ∈ V
∀c ∈ R : c · v ∈ V
0 =
����
0 ·v ∈ K · v
∈K
av + bv = (a + b)v ∈ K · v
c · (a · v) = (ca) · v ∈ K · v
K K
K {0} K
n ≥ 2 K n
U �= {0} K
c ∈ U c �= 0
d ∈ K
d = dc −1 c ∈ U
K ⊂ U K = U
a �= b Ua ∩ Ub = {0}
Ua := K · (1,a,0,...) �= {0}
Ub := K · (1,b,0,...) �= {0}
c(1,a,0,...)=d(1,b,0,...)
K

�
i∈I Ui
�
i∈I Ui
(c, ca, 0,...) = (d, db, 0,...)
c = d
ca = db = cb
c (b − a)
� �� �
�=0
= 0
c = 0
d = 0
(Ui)i∈I
U1 ∪ U2
U1 + U2 := {u1 + u2|u1 ∈ U1,u2 ∈ U2}
U1 + U2
Ui ⊂ V
∀i ∈ I : 0 ∈ Ui
v ∈ �
Ui ⇐⇒ ∀i ∈ I : v ∈ Ui
i∈I
�
i∈I Ui ⊂ V
c ∈ K v, w ∈ �
i∈I Ui
Ui
�
i∈I Ui
V = R 2
0 ∈ �
i∈I
Ui
∀i ∈ I : v, w ∈ Ui
U1 ∪ U2
cv, v + w ∈ Ui
cv, v + w ∈ �
i∈I
� �
1
U1 := R ·
0
� �
0
U2 := R ·
1
Ui
Ui
Ui
Ui

� 0
1
� � � 1
, 0 ∈ U1 ∪ U2
� �
1
+
0
� �
0
=
1
U1 + U2
u = u1 + u2,v = v1 + v2 ∈ U1 + U2
U1 + U2
U1 ⊂ U1 + U2
U2 ⊂ U1 + U2
U1 + U2
� �
1
�∈ U1 ∪ U2
1
0 = 0+0 ∈ U1 + U2
u + v = u1 + u2 + v1 + v2
= (u1 + v1) +(u2 + v2) ∈ U1 + U2
� �� � � �� �
∈U1
∈U1
∈U2
∈U2
c(u1 + u2) = cu1 + cu2 ∈ U1 + U2
���� ����
u1 + u2
U1 ∪ U2
u1 ∈ U1
u2 ∈ U2
u1 = u1 +0∈ U1 + U2
u2 =0+u2 ∈ U1 + U2
W ⊂ V U1 U2
U1 + U2 ⊂ W

v1,...,vn
w ∈ V
v1,...,vn
v1,...,vn
v1,...,vn
K v1,...,vn ∈ V
n�
aivi ∈ V ai ∈ K
i=1
v1,...,vn
∀w ∈ V ∃a1,...,an ∈ K : w =
w = � n
i=1 aivi
w =
n�
i=1
K · v
aivi =
n�
i=1
(u1,...,un) =
bivi
� n�
i=1
⇐⇒
⇐⇒
n�
i=1
aivi
⇐⇒
1 ≤ i ≤ n : ai = bi,
aiui
�
�
�
�
� ai
�
∈ K
u1,...,un
0 ∈ U U = Lin(u1,...,un) a1 = ...= an =0
u + w ∈ U
n�
n�
n�
aiui + biui = (ai + bi)ui ∈ U
i=1
i=1
i=1

cu ∈ U c ∈ K
v1,...,vn
⇒
c
n�
i=1
aiui =
n�
(cai)ui ∈ U
i=1
i) v1,...,vn
n�
ii) aivi =0 ∀1 ≤ i ≤ n : ai =0
iii)
i=1
vk vi
�n i=1 0 · vi =0
n�
0 · vi =0=
i=1
⇒ vj = � n
i=1,i�=j aivi
aj = −1
⇒
aj − bj �= 0
n�
i=1
∀1 ≤ i ≤ n : ai =0
n�
i=1
aivi =0
ai =0 1 ≤ i ≤ n
n�
i=1
−1 =aj =0
aivi =
n�
i=1
bivi
n�
(ai − bi)vi =0
i=1
vj = −1
aj − bj
n�
i=1,i�=j
vi
aivi
(ai − bi)vi

e1,...,en
Kn ⎛
⎜
e1 = ⎜
⎝
1
0
0
⎞ ⎛
0
⎟ ⎜
⎟ ⎜ 1
⎟ ⎜
,e2 ⎟ = ⎜ 0
⎟ ⎜
⎠ ⎝
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ,...,en = ⎜
⎟ ⎜
⎠ ⎜
⎝
⎞
0
⎟
0 ⎠
1
⎛
⎜
a = ⎝
a1
an
a ∈ Kn ⎞
⎟
⎠ =
⎛
⎜
a1 ⎜
⎝
1
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ + ...+ an ⎜
⎠ ⎝
0
0
⎞
⎟
⎠
0
1
= a1e1 + ...+ anen
=
n�
i=1
aiei
�n i=1 aiei =0
⎛ ⎞ ⎛
⎜
⎝
∀1 ≤ i ≤ n : ai =0
v1,...,vn
v1,...,vn
vi
v1,...,vk
a1
an
⎟ ⎜
⎠ = ⎝
v1,...,vn
vi = vj i �= j v1,...,vn
vi =0 v1,...,vn
s : {1,...,n}→{1,...,n}
vi
⎞
0
⎟
⎠
0
n�
as(i)vs(i) =0
i=1
∀1 ≤ i ≤ n : a s(i) =0
v1,...,vn,v

∀1 ≤ i ≤ n : ai =0
�k i=1 aivi =0 ak+1 = ...= an =0
v1,...,vn
v1,...,vk
w = � n
i=1 aivi
v1,...,vn,v
v1,...,vn
0=
n�
i=1
aivi
a1 = ...= ak =0
w =
a =0
n�
aivi + av
i=1
ai = 1 = bj
ak = 0 k �= i
bk = 0 k �= j
n�
akvk = vi = vj =
k=1
a1 = ...= an =0
n�
k=1
∀1 ≤ k ≤ n : ak = bk
bi =0�= 1=ai
bkvk
v1,...,vn
vi =0 ai =1 bi =2 ak = bk =1 k �= i
v1,...,vn
n�
k=1
akvk =
n�
k=1
bkvk
ai =1�= 2=bi
ai = bi

v1,...,vn
v1,...,vn
v1,...,vn−1
w = � n
i=1 aivi
v1,...,vn
n−1 �
w = anvn +
n−1 �
= an
i=1
=
i=1
aivi
�
n−1
bivi + aivi
i=1
n−1 �
(anbi + ai)vi
i=1
⇒
∃j : v1,...,vj−1,vj+1,...,vn
⇐ vj
v1,...,vn
v1,...,vn
vi
v1,...,vr
v1,...,vr,wi1 ,...,wim
vj =
⇐⇒
n�
i=1,i�=j
aivi
v1,...vn
⇐⇒
v1,...,vn
w1,...,ws
vn = � n−1
i=1 bivi
wi1,...,wim

wik
v1,...,vr,w1,...,ws
wj v1,...,vr wi
v1,...,vr
bi =0
v1,...,vr,wi1 ,...,wim
r�
k=1
akvk +
m�
j=1
bjwij =0
l ∈{1,...,m} bl �= 0
wil =
v1,...,vr,wi1 ,...,wim
v1,...,vn
⇐ v ∈ V
a =0
v1,...,vn
r� −ak
k=1
bl
r�
k=1
vk +
s�
j=1,j�=l
akvk =0
∀1 ≤ i ≤ r : ai =0
⇐⇒
−bj
bl
wij
w1,...,ws
⇒ v ∈ V v = � n
i=1 aivi
v ∈ V v1,...,vn,v v1,...,vn
n�
aivi + av =0
i=1
� n
i=1 aivi =0
∀1 ≤ i ≤ n : ai =0

v1,...,vn,v
a �= 0
v1,...,vn
a �= 0
v1,...,vn
v1,...,vn−1
a =0
v1,...,vn−1
v =
n�
i=1
ai
a vi
v1,...,vn−1,w
n−1 �
aw +
w =
i=1
n−1 �
i=1
aivi =0
−ai
a vi
w ∈ V
w v1,...,vn−1
n−1 �
i=1
w =
aivi =0
n�
i=1
aivi
1 ≤ i ≤ n − 1:ai =0
an �= 0 w v1,...,vn−1
vn
vi
vn = 1
an
n−1 �
w +
i=1
v1,...,vn−1,w
−ai
v1,...,vn w1 ...,wm n ≤ m
wj
an
vi

l =1 v2,...,vn,w1 ...,wm
wij
wij
l → l +1 w1,...,wl,vl+1,...,vn
wij
wij
l = n w1,...,wn
m>n
n = m
v2,...,vn,wi1,...,wik
v2,...,vn
v1,...,vn
w1,v2,...,vn
wij = w1
w1,...,wl,vl+2,...,vn,wl+1,...,wm
w1,...,wl,vl+2,...,vn,wi1 ,...,wik
wij = wl+1
v2,...,vn
w1,...,wl,vl+2,...,vn
w1,...,wl,vl+1,...,vn
w1,...,wl,vl+2,...,vn
w1,...,wl+1,vl+2 ...,vn
dim V ≤ n ⇐⇒ v1,...,vn+1
⇒ v1,...,vn+1
dim V ≥ n +1 dim V = n
⇐ v1,...,vm m ≥ n +1
dim V ≤ n
U ⊂ V dim V = n

⇒
⇐ u1,...,un
V ⊂ U
U ⊂ V U = V
R 2
� 1
0
� 1
0
� � � 0
,
1
� � � 0
v1,...,vm
v1,...,vm
U1,U2
v1,...vn
1
U = K ·
� �
1
1
v1,...vn
dim U1 +dimU2 =dimU1 ∩ U2 +dim(U1 + U2)
u1,...,ur
U1 ∩U2,U1 +U2
U1 ∩ U2
U1 ∩ U2 ⊂ U1 u1,...,ur U1
u1,...,ur,v1,...,vm
U1 ∩ U2 ⊂ U2 u1,...,ur U2
u1,...,ur,w1,...,wn
u1,...,ur,v1,...,vm,w1,...,wn
x + y ∈ U1 + U2
x + y =
=
r�
i=1
aiui +
m�
j=1
bjvj +
r�
(ai + a ′ m�
i)ui +
i=1
j=1
r�
i=1
bjvj +
x ∈ U1,y ∈ U2
a ′ iui +
n�
k=1
n�
k=1
ckwk
ckwk
dim V = n
v1,...,vn
U1 + U2

�
i=1
x =
aiui +
r�
i=1
m�
j=1
aiui +
bjvj +
m�
j=1
bjvj
� �� �
∈U1
n�
k=1
= −
ckwk =0
n�
k=1
ckwk
� �� �
∈U2
x ∈ U1 ∩ U2
u1 ...,ur U1 ∩ U2 di ∈ K
u1,...,ur,w1,...,wn
x =0
u1,...,ur,v1,...,vm
i=1
x =
r�
i=1
diui
r�
n�
diui = x = −
r�
i=1
diui +
n�
k=1
U2
k=1
ckwk
ckwk =0
di = 0 1 ≤ 1 ≤ r
ck = 0 1 ≤ k ≤ n
0=x =
r�
i=1
aiui +
U1
m�
j=1
bjvj
ai = 0 1 ≤ i ≤ r
bj = 0 1 ≤ j ≤ m
u1,...,ur,v1,...,vm,w1,...,wn
dim(U1 + U2) =r + m + n

dim U1 +dimU2 = r + m + r + n
= dimU1∩U2 +dim(U1 + U2)

� b
a
f : V → W ⇐⇒
1.) ∀v1,v2 ∈ V : f(v1 + v2) =f(v1)+f(v2)
2.) ∀v ∈ V,c ∈ K : f(cv) =cf(v)
lim
n→∞ : { }→R, (an)n ↦→ lim
n→∞ an
d
′
: {f : R → R }→{f : R → R},f ↦→ f
dx
dx : { f :[a, b] → R} →R,f ↦→
lim
n→∞ (an + bn) = lim
n→∞ an + lim
n→∞ bn
lim
n→∞ can = c lim
n→∞ an
� b
(f + g) ′ (x) = f ′ (x)+g ′ (x)
(cf) ′ (x) = cf ′ (x)
(f + g)(x)dx =
a
� b
a
(cf)(x)dx = c
� b
a
� b
a
f(x)+
f(x)dx
� b
a
g(x)
� b
a
f(x)dx

f1,f2 : U → V
f1 + f2 : U → W, u ↦→ f1(u)+f2(u)
af1 : U → W, u ↦→ af1(u)
f : U → V g : V → W
(f1 + f2)(u1 + u2)
(f1 + f2)(cu1)
(af)(u1 + u2)
(af)(cu1)
{f : U → V }
g ◦ f : U → W, u ↦→ g(f(u))
U
g◦f
−→ W
f ↘ ↗ g
V
u1,u2 ∈ U c ∈ K
Def
= f1(u1 + u2)+f2(u1 + u2)
f1,f2
= f1(u1)+f1(u2)+f2(u1)+f2(u2)
Def
= (f1 + f2)(u1)+(f1 + f2)(u2)
Def
= f1(cu1)+f2(cu1)
f1,f2
= cf1(u1)+cf2(u1)
Def
= c(f1 + f2)(u1)
Def
= af(u1 + u2) = af(u1)+af(u2)
Def
= (af)(u1)+(af)(u2)
Def
= af(cu1) =caf(u1) Def
= c(af)(u1)
g( f(u1 + u2) ) = g( f(u1)+f(u2) )
= g(f(u1)) + g(f(u2))
g( f(cu1) ) = g( cf(u1) ) = cg(f(u1))

f(0) = 0
f : V → W
Null(f) :={v ∈ V |f(v) =0}
Bild(f) :={w ∈ W |∃v : f(v) =w}
f : V → W
v ∈ V 0 · v =0∈ V
f(
����
0 )=f(
����
0 ·
����
v ) =
����
0 · f(v) =
���� ����
0
∈V
∈K ∈V
∈K ∈W ∈W
u, v ∈ Null(f) c ∈ K
0 ∈ Null(f) f(0) = 0
cv ∈ Null(f)
f(cv) =cf(v) =c0 =0
u + v ∈ Null(f)
f(u + v) =f(u)+f(v) =0+0=0
w1,w2 ∈ Bild(f) c ∈ K
v1,v2 ∈ V f(v1) =w1 f(v2) =w2
0 ∈ Bild(f) f(0) = 0
cw1 ∈ Bild(f)
w1 + w2 ∈ Bild(f)
cw1 = cf(v1) =f( cv1 ) ∈ W
����
∈V
w1 + w2 = f(v1)+f(v2) =f(v1 + v2)
∈ Bild(f)
� �� �
∈V

f : V → W
Null(f) ={0}
f(x) =0 x =0
f(x) =f(y) ⇐⇒ x − y ∈ Null(f)
f(x) =f(y) ⇐⇒ f(x) − f(y) =0
⇒ f(0) = 0
⇒ f(x) =0 x =0
⇒
⇐⇒ f(x − y) =0
Def
⇐⇒ x − y ∈ Nullf
Null(f) ={x ∈ V : f(x) =0} = {0}
f(x) =f(y)
1.)
⇐⇒ x − y ∈ Null(f) ={0}
⇐⇒ x − y =0
⇐⇒ x = y
f : V → W
f −1 : W → V
g : W → U g ◦ f : V → U
(g ◦ f) −1 = f −1 ◦ g −1
w1,w2 ∈ W, c1,c2 ∈ K.
f(c1f −1 (w1)+c2f −1 (w2)) = c1f(f −1 (w1)) + c2f(f −1 (w2))
f −1
= c1w1 + c2w2
c1f −1 (w1)+c2f −1 (w2) =f −1 (c1w1 + c2w2)

g ◦ f
g ◦ f
f −1 ◦ g −1 ◦ g ◦ f = f −1 ◦ f = idV
g ◦ f ◦ f −1 ◦ g −1 = g ◦ g −1 = idU
(g ◦ f) −1 = f −1 ◦ g −1
f : V → W dim V = n
Null(f)
dim Null(f) ≤ n
dim Bild(f) ≤ n
dim Null(f)+dimBild(f) = dimV
dim Null(f) ≤ dim V = n
v1 ...,vs Null(f) v1,...,vn
f(vs+1),...,f(vn)
v = �n i=1 aivi
v1,...,vs ∈ Null(f)
∈ V
f(v) =
�
n�
f
�
=
n�
aif(vi)
=
=
s�
i=1
n�
i=1
i=s+1
f
aivi
ai f(vi)
n�
i=s+1
� n�
� �� �
=0
+
aif(vi)
i=s+1
n�
i=s+1
aif(vi) =0
aivi
�
=0
i=1
aif(vi)

v1 ...,vs
v1,...,vn
Bild(f) =W
n�
i=s+1
f(vs+1),...,f(vn)
⇐⇒
n�
i=s+1
n�
i=s+1
aivi +
aivi ∈
aivi =
s�
i=1
bivi
bi
s�
(−bi)vi =0
i=1
bi = 0 1 ≤ i ≤ s
ai = 0 s +1≤ i ≤ n
f : V → W dim V =dimW = n
⇐⇒ Null(f) ={0}
f
{0}⊂Null(f)
⇐⇒ dim Null(f) =dim{0} =0
dim Bild(f)+dim Null(f)=dim V
⇐⇒ dim Bild(f) =dimV − dim Null(f) =dimW
⇒
Bild(f)⊂W
⇐⇒ Bild(f) =W
⇒ Bild(f) =W
⇐⇒ Bild(f) =V

f
v1,...,vn
f : {v1,...,vn} →{w1,...,wn},vi ↦→ wi
⇒ v = �n i=1 aivi
�
f(v) =f
� n�
i=1
aivi +
� n�
i=1
aivi
f : V → W,
f : V → W
linear
=
n�
i=1
u = �n i=1 aivi
n�
�
=
i=1
�
n�
f c
bivi
aivi
i=1
�
=
= f
= c
= c
n�
i=1
aivi ↦→
aif(vi) Vor.
=
n�
i=1
w1,...,wn ∈ W
aiwi
w = �n i=1 bivi
n�
(ai + bi)f(vi)
i=1
n�
i=1
= cf
aiwi +
� n�
i=1
aivi
n�
aif(vi)
i=1
n�
i=1
⇐ wi = f(vi) f : V → W
v1,...,vn
aiwi
� n�
i=1
aivi
n�
i=1
n�
i=1
biwi
aiwi
� �
n�
+ f
�
f : V → W
f(v1),...,f(vn) ⇒
i=1
bivi
�

v1,...,vn
f(v1),...,f(vn) ⇐
v1,...,vn
f(v1),...,f(vn) ⇐⇒
vi
v = �n i=1 aivi
n�
aif(vi) =
�
n�
f
i=1
i=1
f(vi)
dim V =dimW = n
f(v) =0
�
= f(v) =0
aivi
∀1 ≤ i ≤ n : ai =0
v =0 Null(f) ={0}
�n i=1 aif(vi) =0
�
n�
f
i=1
aivi
�
=
n�
aif(vi) =0
i=1
n�
aivi ∈ Null(f) = {0}
i=1
f : V → W
n�
aivi = 0
i=1
∀1 ≤ i ≤ n : ai =0
⇐⇒
v1,...,vn V ⇐⇒ f(v1),...,f(vn)

⇒
w ∈ W
f(v1),...,f(vn)
Bild(f) =W v ∈ V f(v) =w
v1,...,vn
ai ∈ K, 1 ≤ i ≤ n
f(v) =f
v =
� n�
i=1
n�
i=1
aivi
aivi
�
=
n�
aif(vi)
f(v1),...,f(vn)
⇐ f −1 : W → V
f(v1),...,f(vn) f −1 (f(v1)),...,f −1 (f(vn))
f −1 (f(vi)) = vi
⇒
v1,...,vn
i=1
dim V =dimW = n
f : V → W ⇐⇒
v1,...,vn V ⇐⇒ f(v1),...,f(vn) W
⇐ v1,...,vn w1,...,wn
f : {v1,...,vn} →{w1,...,wn},vi → wi
f : V → W,
n�
i=1
aivi ↦→
n�
i=1
dim V = n =dimW
aiwi
f(v1),...,f(vn) w1,...,wn
dim V =dimW = n

f : V → W
K n
(a1,...,an)
⎛
⎜
⎝
a1
an
⎞
⎟
⎠
m × n K K
⎛
⎜
Ab := ⎝
⎛
⎜
⎝
K n
a11 ··· a1n
am1 ··· amn
aij
⎛
⎜
(a1,...,an) ⎝
a11 ··· a1n
am1 ··· amn
(Ab)i =
b1
bn
⎞
⎟
⎠
⎞
n�
⎟
⎠ :=
i=1
aibi
m×n b ∈ K n
n�
j=1
⎞ ⎛
⎟ ⎜
⎠ ⎝
b1
bn
⎞ ⎛
⎟ ⎜
⎠ := ⎝
aijbj =(ai1,...,ain)
� n
j=1 a1jbj
� n
j=1 amjbj
⎛
⎜
⎝
b1
bn
⎞
⎟
⎠
⎞
⎟
⎠ ∈ K m

f : Kn → Km e1,...en e ′ 1,...,e ′ m
Kn Km ⎛
⎜
A := ⎝
b ∈ K n
a11 ··· a1n
am1 ··· amn
b = � n
j=1 bjej ∈ K n
⎛
⎜
b = ⎝
b1
bn
f(b) Def
=
⎛
n�
f ⎝
Def
= b1
K n
=
Def
=
f(b) =Ab
f : K n → K m
⎞
⎟
⎠ =(f(e1),...,f(en))
⎛
⎞
⎜
⎟ ⎜
⎠ = b1 ⎜
⎝
1
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ + ...+ bn ⎜
⎠ ⎝
⎞
0
⎟
0 ⎠
0
1
bjej
j=1
⎛
⎜
⎝
a11
am1
⎞
⎠ linear
=
n�
bjf(ej)
j=1
⎞ ⎛
⎟ ⎜
⎠ + ...+ bn ⎝
⎛
a11b1 + ...+ a1nbn
⎜
⎝
am1b1 + ...+ amnbn
⎛
⎞ ⎛
a11 ··· a1n
⎜
⎟ ⎜
⎝
⎠ ⎝
am1 ··· amn
a1n
amn
⎞ ⎛
⎟ ⎜
⎠ = ⎝
b1
bn
⎞
⎞
⎟
⎠
⎟
⎠ = Ab
� n
j=1 a1jbj
� n
j=1 amjbj
⎞
⎟
⎠

l × m m × n
B =(b1,...,bn) l × n C := AB
e1,...,en
Abi
AB = (Ab1,...,Abn)
⎛ �m j=1
⎜
= ⎝
a1jbj1 ··· �m j=1 a1jbjn
�m j=1 aljbj1 ··· �m j=1 aljbjn
⎞
⎟
⎠
f ◦ g : K n → K l
g : K n → K m f : K m → K l
(f ◦ g)(ei) =
Kn f◦g
−→ l K
g ↘
K
↗ f
m
⎛
⎜
⎝
AB = f ◦ g
� m
i=1 a1jbji
� m
i=1 aljbji
⎞
⎟
⎠
f ◦ g(ei) = f(g(ei)) = A(Bei) =Abi
⎛
⎞ ⎛
a11 ··· a1m
⎜
⎟ ⎜
= ⎝
⎠ ⎝
=
⎛
⎜
⎝
al1 ··· alm
� m
j=1 a1jbji
� m
j=1 anjbji
f ◦ g AB ei
Kn AB = f ◦ g
⎞
⎟
⎠
b1i
bmi
⎞
⎟
⎠

dim V = n B = v1,...,vn B ′ = v ′ 1,...,v ′ n
e1,...,en
K n
pB : {e1,...,en} →V,ei ↦→ vi
p ′ B : {e1,...,en} →V,ei ↦→ v ′ i
pB p ′ B v1,...,vn v ′ 1,...,v ′ n
v1,...,vn
v ′ j
v ′ j =
n�
i=1
sijvi
s : {v ′ 1,...,v ′ n}→V,v ′ j ↦→
1 ≤ j ≤ n
⎛
⎜
S = ⎝
s : V → V
s11 ··· s1n
sn1 ··· snn
pB ◦ S = pB ′
K n
⎜
pB ◦ S(ej) = pB ⎝
S
−→ K n
n�
i=1
⎞
⎟
⎠
sijvi
pB ′ ↘ ↙ pB
(v ′ 1,...,v ′ n) V (v1,...,vn)
linear
=
n�
⎛
s1j
snj
i=1
= v ′ j = pB ′(ej)
⎞
⎟
⎠ = pB(s1je1 + ...+ snjen)
sijpB(ei) Def
=
n�
i=1
sijvi

e1,...,en
pB
p −1
C
p ′ B
pB ◦ S = pB ′
S = p −1
B
◦ pB ′
f : V → W
dim V = n dim W = m
B = v1,...,vn C = w1,...,wm f : V → W
p −1
C ◦ f ◦ pB : K n → V → W → K m
K n
MB,C(f)
p −1
C ◦f◦pB
−→ K m
pB ↓ ↓ pC
(v1,...,vn) V
f
−→ W (w1,...,wn)
◦ f ◦ pB
f : V → W A = MB,C(f)
v1,...,vn w1,...,wm A ′ = MB ′ ,C ′(f)
v ′ 1,...,v ′ n
w ′ 1,...,w ′ m
A ′ = T −1 AS
(v ′ 1,...,v ′ f
n) V −→ W (w ′ 1,...,w ′ n)
p ′ B ↑ ↑ p′ C
Kn A′ =MB ′ ,C ′ (f)
−→ Km S ↓ ↓ T
Kn A=MB,C(f)
−→ Km pB ↓ ↓ pC
(v1,...,vn) V
f
−→ W (w1,...,wn)

MB ′ ,C
′(f) = p−1
C ′ ◦ f ◦ pB ′
= (pC◦T ) −1 ◦ f ◦ pB ◦ S
= T −1 ◦ p −1
C ◦ f ◦ pB ◦ S
= T −1 ◦ MB,C(f) ◦ S

(Ab) T = b T A T
(AB) T = B T A T
(Ab) T =
m × n
A T
a T ij := aji
m × n n × l b ∈ K n
=
⎛
⎜
⎝
� n
j=1 a1jbj
� n
j=1 amjbj
j=1
⎞
⎟
⎠
T
⎛
n�
n�
⎝ a1jbj,...,
⎛
⎜
= (b1,...,bn) ⎝
= b T A T
j=1
amjbj
⎞
⎠
a11 ... am1
a1n ... amn
⎞
⎟
⎠
(AB) T ⎛ �n j=1
⎜
= ⎝
a11bj1 ... �n j=1 a1jbjl
�n j=1 amjbj1 ... �n j=1 amjbjl
⎞T
⎟
⎠
⎛ �n j=1
⎜
= ⎝
a1jbj1 ... �n j=1 amjbj1
�n j=1 a1jbjl ... �n j=1 amjbjl
⎞
⎟
⎠
⎛
⎞ ⎛
⎞
b11 ... bn1 a11 ... am1
⎜
⎟ ⎜
⎟
= ⎝
⎠ ⎝
⎠
= B T A T
b1l ... bnl
a1n ... amn

Tij =
Mi,c =
⎛
1
⎜
⎝
⎛
1
⎜
⎝
Eij,c =
1m
1m
0 ··· 1
1 ··· 0
1
c
⎛
1 c
⎜
⎝
1.) T T ij = Tij
2.) M T i,c = Mi,c
1
1
⎞
⎟
⎠
1
1
⎞
⎟
⎠
⎞
⎟
⎠
(i, i)
1m i �= j (i, j)
3.) E T ij,c = Eji,c
4.) T −1
ij = Tij
5.) M −1
i,c = Mi, 1
c
6.) (Eij,c) −1 = Eij,−c
c �= 0

Eij,c
E T ij,c
i �= j
Tij
Mi,c
akk = 1 1 ≤ k ≤ n
aij = c
akl = 0 k �= l (k, l) �= (i, j)
akk = 1 1 ≤ k ≤ n
aji = c
alk = 0 k �= l (k, l) �= (i, j)
Eji,c
TijTij(ei) = Tij(ej) =ei
TijTij(ej) = Tij(ei) =ej
TijTij(ek) = Tij(ek) =ek i �= k �= j
Mi,cM i, 1
c (ei) = Mi,c
TijTij =1
�
1
c ei
�
= 1
c Mi,c(ei) = 1
c cei = ei
Mi,cM 1 i, c (ek) = Mi,c(ek) =ek k �= i
M 1 i, c Mi,c(ei) = M 1 i, c (cei) =c 1
c ei = ei
M 1 i, c Mi,c(ek) = M 1 i, c ek = ek k �= i
Mi,c · M 1 i, c =1=Mi, 1 · Mi,c
c
Eij,cEij,−c(ej) = Eij,c(−cei + ej)
= −cEij,c(ei)+Eij,c(ej)
= −cei + cei + ej = ej
Eij,−cEij,c(ej) = Eij,−c(cei + ej)
= cEij,−c(ei)+Eij,−c(ej)
= cei − cei + ej = ej

Tija1 =
k �= j
i

T T ij = Tij
ATij
Mi,ca1 =
⎛
1
⎜
⎝
Mi,c = M T i,c
AMi,c
Eij,ca1 =
A T
ET ij,c = Eji,c
AEij,c
A T
T T ij A T = TijA T
ATij =(A T ) T (T T ij ) T =(T T ij A T ) T
1
c
1
1
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
M T i,cA T = Mi,cA T
a1,1
ai−1,1
ai,1
ai+1,1
an,1
AMi,c =(A T ) T (M T i,c) T =(M T i,cA T ) T
⎛
1 c
⎜
⎝
1
⎞
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎝
⎠
a1,1
ai,1
aj,1
an,1
E T ij,cA T = Eji,cA T
⎞
⎟
⎠ =
⎛
⎜
⎝
A T
AEij,c =(A T ) T (E T ij,c) T =(E T ij,cA T ) T
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ = ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
a1,1
a1,1
ai−1,1
cai,1
ai+1,1
an,1
ai,1 + caj,1
aj,1
an,1
⎞
⎟
⎠
⎞
⎟
⎠

A ′ ⎛
⎜
= ⎜
⎝
0 0
Eij,c
0 ...0 a ′ 1j1 ... ∗ 0 ∗ ...∗ 0 ∗ ∗
0 ...0 0 ... 0 a ′ 2j2 ∗ ...∗ 0
0 ∗ ∗
0 ...0 0 0 ... 0 ... a ′ rjr ... a ′ rm
j1
a1,j1 =0 E1j,1
A ′ = E1j,1A =
A ′ = Es ...E1A
⎛
0
⎜
A = ⎝
··· 0 ∗ ···
⎞
∗
⎟
⎠
0 ··· 0 ∗ ··· ∗
a ′ 1,j1
a ′ 1,j1
⎛
⎜
⎝
∃j : aj,j1 �=0
=0+1· aj,j1 �=0
0 ··· 0 a ′ 1,j1
0 0
�=0 ···
⎞
∗
⎟
⎠
0 ··· 0 ∗ ··· ∗
∀2 ≤ i ≤ n : a ′ i,j1 =0
Ei1,ci
ci = − a′ i,j1
a ′ 1,j1
n × m
⎞
⎟
⎠

a ′′
i,j1 = a′ i,j1 + −a′ i,j1
a ′ 1,j1
2 ≤ i ≤ n
A1 = En1,cn ...E21,c2E1j,1A =
a2,j2 =0 E2j,1
a ′ 1,j1 =0
⎛
⎜
⎝
0
0
···
···
0
0
a1,j1
0
···
···
∗
0
⎞
∗
∗ ⎟
⎠
0 ··· 0 0 ··· 0 ∗
∃j ≥ 2:aj,j2 �=0
a ′ 2,j2 =0+1· aj,j2 �= 0
A ′ = E2j,1A1
⎛
0 ··· 0 a1,j1 ··· ∗ ∗
⎜ 0 ··· 0 0 ··· 0 a ′ 2,j2 �=0
=
⎜
⎝
0 ··· 0 0 ··· 0 ∗
a ′ 2,j2
Ei2,ci
ci = − a′ i,j2
a ′ 2,j2
a ′′
i,j2 = a′ i,j2 + −a′ i,j2
a ′ 2,j2
a ′ 2,j2 =0
1 ≤ k ≤ j2 − 1 ai,k =0 2 ≤ i ≤ n 3 ≤ i ≤ n
a ′′
i,k = a ′ i,k
����
=0
+ −ai,j2
a2,j2
a ′ 2,k =0
����
=0
⎞
⎟
⎠

j2 − 1
3 ≤ i ≤ n
A2 = En2,cn ...E32,c2E2j,1A1 ⎛
0 ...0 a ′ ... ∗ ∗ ∗...∗ ∗
1j1
=
⎜
⎝
A ′ ⎛
⎜
= ⎜
⎝
0 0
⎛
A ′′ ⎜
= ⎜
⎝
0 ...0 0 ... 0 a ′ 2j2 ∗ ...∗
0 ...0 0 ... 0 0 0...0 ∗
0 ...0 a ′ 1j1 ... ∗ ∗ ∗...∗ ∗ ∗ ∗
0 ...0 0 ... 0 a ′ 2j2 ∗ ...∗
⎞
⎟
⎠
∗ ∗ ∗
0 ...0 0 0 ... 0 ... a ′ rjr ... a ′ rm
ak,ji
1 ≤ k ≤ i − 1
Ekji,ck
1 ≤ l

⎛
A ′′′ ⎜
= ⎜
⎝
j3,...,jr
0 ...0 a ′ 1j1 ... ∗ 0 ∗ ...∗ 0 ∗ ∗
0 ...0 0 ... 0 a ′ 2j2 ∗ ...∗
0 ∗ ∗
0 ...0 0 0 ... 0 ... a ′ rjr ... a ′ rm
0 0
s1,...,sn
z ′ 1,...,z ′ r
n × n
A ′ ⎛
⎜
= ⎝
0
z1,...,zm
Lin(z1,...,zm)
m × n
Lin(s1,...,sn)
Lin(z ′ 1,...,z ′ r)=Lin(z1,...,zn)
a ′ 11
0
0 a ′ nn
⇐⇒ dim Lin(z1,...,zn) =n
j1,...,jr
s ′ j1 ,...,s′ jr
sj1 ,...,sjr
s ′ j1 ,...,s′ jr
j1,...,jr
⎞
m × n
Lin(z1,...,zn)
⎟
⎠ ∀1 ≤ i ≤ n : a ′ ii �= 0
Lin(s ′ 1,...,s ′ n)
⇐⇒
Lin(s1,...,sm)
dim Lin(z1,...,zn) =dimLin(s1,...,sm)
⎞
⎟
⎠

n�
k=1
A ′ = Eij,cA z ′ i = zi + czj
� n
k=1 bkzk ∈ Lin(z1,...,zn)
bkzk =
=
n�
k=1,i�=k�=j
n�
k=1,i�=k�=j
∈ Lin(z ′ 1,...,z ′ n)
�n k=1 b′ kz′ k ∈ Lin(z′ 1,...,z ′ n)
n�
k=1
b ′ kz ′ k
=
=
z ′ r+1 = ...= z ′ n =0
z ′ 1,...,z ′ r
� r
k=1 bkz ′ k =0
n�
k=1,i�=k�=j
n�
k=1,i�=k�=j
bkzk + bjzj + bi (zi + czj) −biczj
∈ Lin(z1,...,zn)
� �� �
=z ′ i
bkz ′ k +(bj − cbi)z ′ j + biz ′ i
b ′ kzk + b ′ i(zi + czj)+b ′ jz ′ j
bkzk + b ′ izi +(b ′ j + cb ′ i)zj
Lin(z ′ 1,...,z ′ r) = Lin(z ′ 1,...,z ′ n)
= Lin(z1,...,zn)
(0,...,0,b1a ′ 1j1 , ∗,...,∗,b2a ′ 2j2 , ∗,...,∗,bra ′ rjr , ∗,...)
= (0,...,0, 0, 0,...,0, 0, 0,...,0, 0)
a ′ kjk
1 ≤ k ≤ r
�=0 1 ≤ k ≤ r
z ′ 1,...,z ′ r
bka ′ kjk =0
bk =0
Lin(z1,...,zn)
Lin(z ′ 1,...,z ′ r)
dim Lin(z1,...,zn) =n ⇐⇒ dim Lin(z ′ 1,...,z ′ r)=n
⇐⇒ r = n
⇐⇒ ∀1 ≤ i ≤ n : aii �= 0

sj1 ,...,sjr
� r
j=1 bis ′ ji =0
⎛
0
⎞
⎜ 0
⎜ 0
⎜
⎝
0
⎟ =0=
⎟
⎠
a ′′ �=0 1 ≤ i ≤ r
i,ji
s ′ k
s ′ j1 ,...,s′ jr
E1,...,Em
sj1 ,...,sjr
r�
r�
i=1
bisji =
bi =0
ek = 1
ak,jk
s ′ jk
r�
⎛
⎜
⎝
b1a ′′
1j1
bra ′′
rjr
0
s ′ k = a
i=1
′ a
ikek =
i=1
′ ik
a ′ s
k,jk
′ jk
0
⎞
⎟
⎠
Lin(s ′ 1,...,s ′ m)
E := Em ◦ ...◦ E1
⇐⇒ Esj1 ,...,Esjr
Lin(z1,...,zn) b)
= r = Lin(sj1 ,...,sjr )
g)
= Lin(s1,...,sn)
dim Lin(z1,...,zn) =dimLin(s1,...,sm)

K n
⇐
⇒
a11x1 + ...+ a1nxn = b1
am1x1 + ...+ amnxn = bm
Ax = b
Ax = b
Ax =0
Ax = b ⇐⇒ CAx = Cb
Null(A) = {x ∈ R n : Ax =0}
= { Ax =0}
x Ax = b ⇒ x CAx = Cb
C −1
⇒ x Ax = b
y Ay = b
Ax = b
Ax = b ⇐⇒ x − y ∈ Null(A)
y + Null(A)
A(x − y) =Ax − Ay = b − b =0
Ax = A(x − y)+Ay =0+b = b

Eij,c
A ′ = Es ...E1A
⎛
0 ...0 a ′ ... ∗ 0 ∗ ...∗ 0 ∗ ∗
1j1
=
T1j1 ...Trjr
⎜
⎝
0 ...0 0 ... 0 a ′ 2j2 ∗ ...∗
0 0 ∗ ∗
0 ...0 0 0 ... 0 ... a ′ rjr ... a ′ rn
A ′′ = M i, 1
a r,jr
=
M i, 1
a i,ji
...M 1 i, A
a1,j1 � �� �
′
=M
a ′ i,ji
0 0
=1 1 ≤ i ≤ r
⎛
0 ...0 1 ... ∗ 0 ∗ ...∗ 0 ∗
⎞
∗
⎜
0
⎜ 0
⎜
⎝
...0
...0
0
0
...
0
0
...
1
0
∗ ...∗
...
0
1
0
∗
...
⎟
∗ ⎟
∗ ⎟
0 ⎟
⎠
Tiji
S = MEs ...,E1
1 ≤ i ≤ r T =
A ′′′ ⎛
1 0 ∗ ... ∗
⎞
⎜
= SAT = ⎜
⎝
⎟
⎠
0 1 a ′ r,r+1 ... a ′ r,n
0 0 0 0 0
Ax = b ⇐⇒
x ∈ TSb+ Lin(Tv1,...,Tvr)
Ax = b ⇐⇒ SAx = Sb ⇐⇒ SAT(T −1 x)=Sb
⎞
⎟
⎠

′ = Sb y = T −1 x
Ax = b ⇐⇒ SATy = b ′
⎛
1
⎜
SATy = ⎜
⎝ 0
0 0
0
1
0
∗
ar,r+1
0
...
...
0
∗
ar,n
⎞
⎟
⎠ y ! ⎛
⎜
= ⎜
⎝
b ′ k
b ′ k
�=0 k>r
(SAT)k =
=0 k>r
SATy = b ′ b ′
n�
⎛
⎜
⎝
1 0 ∗ ... ∗
r +1≤ j ≤ n
0 1 a ′ r,r+1 ... a ′ r,n
0 0 0 0 0
aki yi =0�= b
����
i=1
=0
′ k
⎛
⎞
⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎜
⎝
vj =(−a1j,...,−arj, 0 ...,0, 1
����
b ′ 1
b ′ r
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ = ⎜
⎟ ⎜
⎠ ⎝
SATy =0
b ′ 1
b ′ r
b ′ r+1
b ′ 1
b ′ r
0
, 0,...,0)T
⎞
⎟
⎠
⎞
⎟
⎠

⎜
⎛
⎞ ⎜
1 0 a1,r+1 ... a1,n ⎜
⎜
⎟ ⎜
⎜
⎟ ⎜
SATvj = ⎜
⎟ ⎜
⎝ 0 1 ar,r+1 ... ar,n ⎠ ⎜
0 0 0 0 0 ⎜
⎝
=
⎛
⎜
⎝
−a1j + a1j
−arj + arj
0
vr+1,...,vn
�
n − r
n
j=r+1 bjvj =0
vj
⎞
⎟ =0
⎟
⎠
⎛
−a1j
−arj
0
0
1 ←
0
...
(∗,...,∗,br+1 ...,bn) =(0,...,0, 0,...,0)
bj =0 r +1≤ j ≤ n
dim Null(SAT)=n − dim Bild(SAT)=n − r
Null(SAT)
y = b ′ + Null(SAT)
y = Sb + Lin(vj+1,...,vn)
x = Ty = TSb+ Lin(Tvj+1,...,Tvn)
SAx = SATSb + SAT
= SATb ′ +
= b ′ = Sb
Ax = b
n�
k=j+1
n�
k=j+1
akvk
ak SATvk
� �� �
=0
⎞
⎟
⎠

R
c ∈ R,u,v,w ∈ V
〈·, ·〉 : V × V → R
1.) 〈cv, w〉 = c 〈v, w〉
2.) 〈u + v, w〉 = 〈u, w〉 + 〈v, w〉
3.) 〈v, w〉 = 〈w, v〉
4.) 〈v, v〉 > 0 v �= 0
v, w ∈ V ⇐⇒
v1,...,vn
R n
〈vi,vj〉 =
〈v, w〉 =0
⇐⇒
〈v, v〉 =1
⎜
〈a, b〉 := (a1,...,an) ⎝
〈a + a ′ ,b〉 =
=
� 1 i = j
0 i �= j
⎛
b1
bn
⎞
n�
⎟
⎠ =
n�
(ai + a ′ i)bi
i=1
n�
i=1
aibi +
n�
i=1
= 〈a, b〉 + 〈a ′ ,b〉
i=1
a ′ ibi
R
aibi
⇐⇒

n v1,...,vn
n�
n�
〈ca, b〉 = (cai)bi = c aibi = c 〈a, b〉
〈a, a〉 =
〈a, b〉 =
i=1
〈a, a〉 =
n�
i=1
n�
i=1
aibi =
n�
i=1
aiai =
i=1
n�
biai = 〈b, a〉
i=1
n�
|ai| 2 ≥ 0
i=1
|ai| 2 =0 ⇐⇒ ∀i : ai =0 ⇐⇒ a =0
∀a �= 0:〈a, a〉 > 0
R 〈·, ·〉 dim V =
j=1
v =
n�
〈v, vi〉 vi
i=1
v = �n j=1 ajvj
�
n�
�
〈v, vi〉 = ajvj,vi =
w1,...,wn
v1,...,vn
n�
aj 〈vj,vi〉 = ai
j=1
Lin(v1,...,vi) =Lin(w1,...,wi)
w1 :=
v1
�
〈v1,v1〉
�i−1
∀i ≥ 2: ui := vi −
∀i ≥ 2: wi :=
j=1
ui
� 〈ui,ui〉
〈vi,wj〉 wj

i =1:
〈w1,w1〉 =
=
�
v1
�
〈v1,v1〉 ,
�
v1
�
〈v1,v1〉
〈v1,v1〉
� 2 =1
〈v1,v1〉
i − 1 → i v1,...,vn ui �= 0
k 0
wi =
ui
� 〈ui,ui〉
∀k, j ≤ i − 1,k �= j : 〈wj,wk〉 =0
〈ui,wk〉 =
�
i−1
vi −
�
�
〈vi,wj〉 wj,wk
j=1
�i−1
= 〈vi,wk〉− 〈vi,wj〉〈wj,wk〉
j=1
= 〈vi,wk〉−〈vi,wk〉 =0
〈wi,wk〉 =
1
�
〈ui,ui〉 〈ui,wk〉 =0
〈wi,wi〉 =
=
wj = � n
i=1,i�=j aiwi
0=〈wj,wk〉 =
�
ui
�
〈ui,ui〉 ,
�
ui
�
〈ui,ui〉
〈ui,ui〉
� 2 =1
〈ui,ui〉
n�
i=1,i�=j
∀k �= j
〈wi,wk〉 = ak

wj =0 〈wj,wj〉 =1
i =1
i − 1 → i
v1
� ∈ Lin(v1)
〈v1,v1〉
v1 = w1 · � 〈v1,v1〉 ∈Lin(w1)
w1 =
∀k

f(e1),...,f(en)
〈f(v),f(w)〉 V
f : {e1,...,en} →V,ei ↦→ vi
=
=
Def
=
=
f : R n → V
� �
n�
f
n�
i=1
i=1 j=1
n�
n�
i=1 j=1
n�
i=1
n�
aiei
� ⎛
n�
,f⎝
bjej
j=1
aibj 〈f(ei),f(ej)〉 V
aibj 〈vi,vj〉 V
aibi = 〈a, b〉 R n
⎞�
⎠
V

|x| ≥ 0
|cx| = |c||x|
|x + y| ≤ |x| + |y|
|x| =0 ⇒ x =0
K
x+y cx
�·�: V → [0, ∞),x↦→� x �
1.) � x �= 0⇒ x =0
2.) ∀c ∈ K ∀x ∈ V : � cx �= |c| �x �
3.) ∀x, y ∈ V : � x + y �≤� x � + � y �
(V,�·�)
(V,� ·�)
(R, |·|), (C, |·|)
∀a, b ∈ R : a + b, ab ∈ R
R
R R ∀x, y, c ∈ R
|x| ≥ 0
|x| =0 ⇒ x =0
|cx| = |c||x|
|x + y| ≤ |x| + |y|
|·|

(R, |·|)
�·�2
x, y ∈ R n
(C, |·|)
Rn �·�2: R n → [0, ∞),x↦→ � �
�
�
〈x, x〉 = � n �
� x �2= 0 ⇐⇒ x =0
|〈x, y〉|≤�x �2� y �2
|〈x, y〉|=� x �2� y �2 ⇐⇒
� n
i=1 x2 i
√ ·
x =0 y =0
i=1
x 2 i
≥ 0 ≥ 0
x =0 ⇐⇒ ∀1 ≤ i ≤ n : xi =0
n�
⇐⇒ 〈x, x〉 =
i=1
x 2 i =0
⇐⇒ � 〈x, x〉 =� x �2= 0
|〈x, y〉|=0≤� x �2� y �2
x, y �= 0 � x �2> 0 � y �2> 0 t>0
0 ≤
=
�
�
�
1
�tx ±
t y
�2
�
�
�
2
�
tx ± 1 1
y, tx ±
t
= 〈tx, tx〉±
�
tx, 1
t y
t y
�
�
= t 2 � x � 2 2 + 1
t 2 � y �2 2 ±2 〈x, y〉
� � �
1 1 1
± y, tx + y,
t t t y
�

t2 � y �2
=
� x �2
x =0 y =0
x �= 0�= y
0 ≤ 2 � x �2� y �2 ±2 〈x, y〉
|〈x, y〉|= ±〈x, y〉 ≤�x �2� y �2
|〈x, y〉|=0=� x �2 ·�y �2
⇒ 〈x, y〉 =� x �2� y �2 t 2 =
�
�
�
1
�tx −
t y
�
�
�
�
〈x, y〉 = −�x �2� y �2
�
�
�
1
�tx +
t y
�
�
�
�
2
2
2
2
� y �2
� x �2
= t 2 � x � 2 2 + 1
t 2 � y �2 2 −2 〈x, y〉
= 2 � x �2� y �2 −2 � x �2� y �2
= 0
tx − 1
y
t
= 0
tx = 1
t y
t 2 =
� y �2
� x �2
= t 2 � x � 2 2 + 1
t 2 � y �2 2 +2 〈x, y〉
= 2 � x �2� y �2 −2 � x �2� y �2
= 0
tx + 1
y
t
= 0
tx = − 1
t y
x = −1
y
t2

⇐ x = ay
|〈x, y〉| =
=
|a|·|〈x, x〉|
� 〈x, x〉 � a2 =
〈x, x〉
� 〈x, x〉 � 〈ax, ax〉
= � x �2� y �2
� cy �2 = � 〈cy, cy〉 = � c2 〈y, y〉
= |c| � 〈y, y〉 = |c| �y �2
� x + y � 2 2 = � x � 2 2 +2 〈x, y〉 + � y � 2 2
≤ � x � 2 2 +2|〈x, y〉|+ � y � 2 2
b)
≤ � x � 2 2 +2 � x �2� y �2 + � y � 2 2
= (�x�2 + � y �2) 2
K n
�·�1: K n → [0, ∞),x↦→
n�
|xi|
i=1
�·�∞: K n → [0, ∞),x↦→ max{|x1|,...,|xn|}
� x �1= 0
� cx �1 =
i=1
Def
⇐⇒
n�
|xi| =0
i=1
⇐⇒ ∀1 ≤ i ≤ n : |xi| =0
⇐⇒ ∀1 ≤ i ≤ n : xi =0
n�
n�
|cxi| = |c| |xi| = |c| �x �1
i=1
√ ·

� x + y �1 =
� x �∞= 0
� cx �∞ =
n�
|xi + yi| ≤
i=1
= � x �1 + � y �1
n
max
Def
⇐⇒
n�
|xi| +
i=1
n�
|yi|
i=1
n
max
i=1 |xi|
⇐⇒
=0
∀1 ≤ i ≤ n : |xi| =0
⇐⇒ ∀1 ≤ i ≤ n : xi =0
i=1 |cxi| = |c| n
max
i=1 |xi| = |c| �x �∞
|xi| + |yi| ≤
n
max
i=1 (|xi| + |yi|) ≤
n
max
i=1 |xi| + n
max
n
max
i=1 |yi|
i=1 |xi| + n
max
i=1 |yi|
� x + y �∞ =
n
max
i=1 |xi + yi| ≤ n
max
i=1 (|xi| + |yi|)
≤
n
max
i=1 |xi| + n
max
i=1 |yi|
= � x �∞ + � y �∞
a) |xi| ≤�x �2
b) � x �∞≤� x �2≤ √ n � x �∞
c) � x �2≤� x �1
|xi| 2 ≤
n�
|xi| 2
i=1
�
�
�
|xi| ≤�
n �
|xi| 2 =� x �2
i=1

x ↦→ x 2
|xi| 2 ≤
n�
|xi| 2 ≤
i=1
� x �∞= n
max
i=1 |xi| ≤�x �2
n
max
i=1 |xi| 2 =
n�
i=1
� n
max
i=1 |xi|
� x �2 ≤ √ n � x �∞
� x � 2 2=
n�
|xi| 2 �
n�
≤ |xi|
i=1
A : V → W B : W → U
i=1
� x �2≤� x �1
1 ≤ i ≤ n
�
n
max
i=1 |xi|
�2 �2 = n � x � 2 ∞
� 2
=� x � 2 1
A : V → W
� A �V,W= sup
�x�V =1
� Ax �W
∀x ∈ V : � Ax �≤� A �� x �
� AB �V,U≤� A �V,W� B �W,U
A, B : V → W
0, cA, A + B : V → W

� Ax � + � Bx � ≤ sup � Ax � + sup � Bx �
�x�=1
�x�=1
sup (� Ax � + � Bx �) ≤ sup � Ax � + sup � Bx �
�x�=1
�x�=1
�x�=1
x =0
� A � = sup
�x�=1
� Ax �≥ 0
� cA � = sup � cAx �= sup |c| �Ax �
�x�=1
�x�=1
= |c| sup
�x�=1
� Ax �= |c| �A �
� A + B � = sup
�x�=1
� Ax + Bx �
≤ sup (� Ax � + � Bx �)
�x�=1
≤ sup � Ax � + sup � Bx �
�x�=1
�x�=1
= � A � + � B �
� 0 � = sup � 0x �= sup � 0 �= 0
�x�=1
�x�=1
� A �= 0 ⇐⇒ sup
�x�=1
� Ax �= 0
⇐⇒ ∀x :� Ax �= 0
⇐⇒ ∀x : Ax =0
⇐⇒ A ≡ 0
� A · 0 �= 0≤� A �� 0 �
x �= 0 � �
��� x �
�
� x ��
= � x �
� x �

(xk)k
⇐⇒
� Ax � =
=
�
�
�
�
x �
�� x � A �
� x ��
� �
�
� x � �
x �
�A �
� x ��
≤ � x � sup
�y�=1
� Ay �
≤ � x �� A �
� AB � ≤ sup
�x�=1
� ABx �
≤ sup
�x�=1
� A �� Bx �
≤ � A � sup
�x�=1
� Bx �
= � A �� B �
d : X × X → [0, ∞), (x, y) ↦→ d(x, y)
1) d(x, y) ≤ d(x, z)+d(z,y)
2) d(x, y) = 0 ⇐⇒ x = y
3) d(x, y) = d(y, x)
x ∈ X ⇐⇒
∀ε >0 ∃K ∈ N ∀k ≥ K : d(x, xk)

limk→∞ xk = x
(xk)k
⇐⇒
∀ε >0 ∃K ∈ N ∀k, l ≥ K : d(xk,xl) 0
2
k ≥ max(k1,k2)
∃k1 ∀k ≥ k1 : d(xk,x) 0 ∀z ∈ X
( dX(x, z) 0 ∃y
∀ε >0 ∃k0 ∀k ≥ k0 : d(f(xk),f(x))

δ = 1
n
(a1,...,an) ⇐⇒
yn
limn→∞ yn = x limn→∞ f(xn) =f(x)
∃n0 ∈ N ∀n ≥ n0 : d(f(x),f(yn))

k ≥ max n i=1 k0,i
⇒
∀k, l ≥ k0
� x − a �2 ≤ √ n � x − a �∞
= √ n n
max
i=1 |xk,i − ai|
< √ n ε
√ = ε
n
∃k0 ∀k, l ≥ k0 � xk − xl �2< ε
|xk,i − xl,i| ≤�xk − xl �2< ε
⇐ ∀1 ≤ i ≤ n ∃k0,i ∀k, l ≥ k0,i
k, l ≥ max n i=1 k0,i
ai ∈ R
(xk)k
|xk,i − xl,i| < ε
√ n
� xk − xl �2 ≤ √ n � xk − xl �∞
= √ n n
max
i=1 |xk,i − xl,i|
< √ n ε
√ = ε
n
(R n , �·�2)
R n
(xk,i)k
(xk)k
R
a =(a1,...,an)
f =(f1,...fn) :(Rm , �·�2) → (Rn , �·�2) ⇐⇒
∀1 ≤ i ≤ n : fi :(R m , �·�2) → (R, |·|)
f =(f1,...fn) :R m → R n
⇐⇒ lim
k→∞ xk = x lim
k→∞ f(xk) =f(x)
⇐⇒ ∀1 ≤ i ≤ n : lim
k→∞ xk = x lim
k→∞ fi(xk) =fi(x)
⇐⇒ ∀1 ≤ i ≤ n : fi : R m → R

(X, d) (R n , �·�2)
B(x, ε) = {y ∈ X : d(x, y) 0:B(x, ε) ⊂ U
x ∈ X ε
(a, b) ={y ∈ R : a

Uj
Ui
ε>0
x ∈∅
∀x ∈∅∃ε>0:B(x, ε) ⊂∅
B(x, ε) ⊂ X
∀x ∈ X ∃ε >0:B(x, ε) ⊂ X
x ∈ �
i∈I Ui
j ∈ I x ∈ Uj
ε>0
x ∈ � n
i=1 Ui
ε := min n i=1 εi
B(x, ε) ⊂ Uj
B(x, ε) ⊂ Uj ⊂ �
εi
i∈I
B(x, εi) ⊂ Ui
Ui
∀1 ≤ i ≤ n : B(x, ε) ⊂ B(x, εi) ⊂ Ui
B(x, ε) ⊂
n�
i=1
Ui
A ⊂ (X, d) ⇐⇒
A C = X\A
⇐⇒
(xk)k ∀k ∈ N : xk ∈ A limk→∞ xk = x
x ∈ A
A A

⇒ x �∈ A
A C x ∈ A C ε>0
limk→∞ xk = x
xk ∈ A
⇐ A C
ε = 1
x ∈ [a, b]
k
B(x, ε) ⊂ A C
∃k0 ∀k ≥ k0 : d(xk,x) 0 ∃y ∈ A : d(x, y) ≥ ε
yk
limk→∞ yk = x ∀k ∈ N : yk ∈ A
x ∈ A x ∈ A C
a, b ∈ R n
[a, b] :={x∈ R n : ∀1 ≤ i ≤ n : ai ≤ xi ≤ bi}
(a, b) :={x∈ R n : ∀1 ≤ i ≤ n : ai

n
max
i=1 xi ≤� x �2≤ √ n n
max
i=1 xi
x ∈ (−C, C) ⇐⇒ ∀1 ≤ i ≤ n : −C

(f(xk))k
(f(xkj ))j
(xk)k
f : A ⊂ R n → R
c = sup
x∈[a,b]
f(x)
lim
k→∞ f(xk) =c
(d1,...,dn) ∈ A
lim f(xkj )=c
j→∞
A ⊂ R n
�
c = lim f(xkj )=f
j→∞
lim
j→∞ xkj
�
= f(d)
(x1kj ,...,xnkj )

K ⊂ X ⇐⇒
K ⊂ �
Ui ⇒
m�
∃m∈N∃i1,...,im : K ⊂
i∈I
M := max(i1,...,im)
y ∈ K
d(x, 0) < ∞
∀x ∈ X ∃i ∈ N : x ∈ B(0,i)
K ⊂ X =
K ⊂
∞�
B(0,i)
i=1
m�
B(0,ij)
j=1
K ⊂ B(0,M)
x ∈ K C
d(x, y) =0 ⇐⇒ x = y
Ui
∀y ∈ K ∃i ∈ N : d(x, y) > 1
i
�
y ∈ B x, 1
�C
i
j=1
Uij
K ⊂ X

� �
1 1
ε := min ,..., i1 im
A ⊂ �
i∈I Ui
A ⊂ K
∞�
K ⊂
K ⊂
i=1
m�
j=1
�
B x, 1
�C
i
�
B x, 1
�C
ij
K ⊂ B(x, ε) C
B(x, ε) ⊂ B(x, ε) ⊂ K C
K ⊂ X = A C ⊂
∪ A
A C
�
����
∪ Ui
i∈I
� �� �
K ⊂ A C ∪
m�
j=1
A ⊂ K ⊂ A C ∪
A ⊂
m�
j=1
Uij
Uij
n�
j=1
Uij
A ⊂ K

(Ak)k
(A) =inf{c ∈ R |∀x, y ∈ A : d(x, y)

lim
m→∞
Qm
Q0
|I| = ∞
Ui
2 m
Qm
Qm+1 ⊂ Qm
c ∈ [a, b] ⊂ �
i∈I Ui
Ui0
c ∈ Qm
Qm
(Qm) = 1
2
Q0 := [a, b] ⊂ �
i∈I
Ui
Ui
Qm−1 = I1 × ...× Im
I1,...,Im
(Qm−1) = 1
(Qm) = (Q0) lim
m→∞
A ⊂ R n
Qm
ε>0
A ⇐⇒ A
i0
c ∈ Ui0
B(c, ε) ⊂ Ui0
(Qm) < ε
2
x ∈ Qm ⇒ d(x, c) < ε
2
Qm ⊂ B(c, ε) ⊂ Ui0
Ui0
Ui
2 m−1
1
=0
2m−1 (Q0)

⇒
⇐ C>0
A ⊂ [−C, C]
[−C, C] A ⊂ [−C, C]
f −1
x ∈ f −1
f : X → Y U ⊂ Y
f −1 (U) ={x ∈ X : f(x) ∈ U}
f −1 � U C�
� �
�
Ui
=
=
� f −1 (U) � C
�
f −1 (Ui)
i∈I
i∈I
A ⊂ B ⇒ f −1 (A) ⊂ f −1 (B)
f(A) ⊂ B ⇐⇒ A ⊂ f −1 (B)
x ∈ f −1 (U C ) ⇐⇒ f(x) ∈ U C
� �
i∈I
Ui
�
⇐⇒ f(x) �∈ U
⇐⇒ x �∈ f −1 (U)
⇐⇒ x ∈ f −1 (U) C
⇐⇒ f(x) ∈ �
i∈I
Ui
⇐⇒ ∃i0 ∈ I : f(x) ∈ Ui0
⇐⇒ ∃i0 ∈ I : x ∈ f −1 (Ui0 )
⇐⇒ x ∈ �
f −1 (Ui)
i∈I
x ∈ f −1 (A) ⇒ f(x) ∈ A
⇒ f(x) ∈ B
⇒ x ∈ f −1 (B)

⇐
⇒
x ∈ A ⇒ f(x) ∈ f(A)
⇒ f(x) ∈ B
⇒ x ∈ f −1 (B)
x ∈ A ⇒ x ∈ f −1 (B)
⇒ f(x) ∈ B
f : X → Y
⇐⇒ U ⊂ Y f −1 (U)
⇐⇒ A ⊂ Y f −1 (A)
⇒ U ⊂ Y x ∈ f −1 (U) f(x) ∈ U
ε>0 B(f(x),ε) ⊂ U
δ>0
d(z,x) 0
B(x, δ) ⊂ f −1 (B(f(x),ε))
∀ε >0 ∃δ >0 ∀x ∈ X :(d(z,x) 0 ∃δ >0 ∀x ∈ X :(d(z,x)

f(K)
A ⊂ X
Ui
f(K)
(yn)n
f : X → Y K ⊂ X f(K)
K ⊂ f −1
K ⊂
f(K) ⊂
∀A ⊂ X
f(K)
�
f
f(K) ⊂ �
i∈I Ui
� �
�
= �
i∈I
Ui
i∈I
f −1 (Ui)
� �� �
m�
f −1 ⎛
m�
−1
(Uij )=f ⎝
j=1
m�
j=1
Uij
f : K → R
lim
n→∞ yn =supf(x)
x∈K
lim
n→∞ yn ∈ f(K)
lim
n→∞ yn
�
=supf(x)
x∈K
Uij
j=1
⎞
⎠
f(K)
(X, d) x ∈ X
dist(x, A) =inf{d(x, y) :y ∈ A}
dist : X → [0, ∞),x↦→ inf{d(x, y) :y ∈ A}

y ∈ A
d(x, x ′ ) 0: B(q, ε) ⊂ A C
∀y ∈ B(q, ε) :y �∈ A
dist(K, A) =dist(q, A) ≥ ε
A ∩ K = ∅

⇐⇒
�·�2
�·�, �·� ∗
∃c, C > 0 ∀x ∈ V : c � x �≤� x � ∗ ≤ C � x �
c � x �≤� x � ∗ ≤ C � x �
d � x � ∗ ≤� x � ∗∗ ≤ D � x � ∗
�·�∞
c =1 C = √ n
� x �≤� x �≤� x �
c � x �≤� x � ∗ ≤ C � x �
⇒ 1
C � x �∗≤� x �≤ 1
� x �∗
c
�·�2
�
� x �∞≤� x �2≤ √ n � x �∞
⇒ cd � x �≤� x � ∗∗ ≤ CD � x �
K dim V = n

√ · x ↦→ x 2
�
�
� lim
S := {x ∈ K n :� x �2= 1}
� x �2= 1< 2 S ⊂ B(0, 2)
(xn)n
�
�
n→∞ xn�
2
=
�
�
�
� n �
lim
= lim
n→∞
i=1
x2n,i n→∞
�
�
�
� n �
i=1
limn→∞ xn = x
x 2 n,i
= lim
n→∞ � xn �2 =1
� �� �
=1
lim
n→∞ xn ∈ S
c := inf{� x �∗: x ∈ S}
= inf{� x �∗: x ∈ K n , � x �2= 1}
c =0 (xn)n
lim
n→∞ � xn �∗= 0
K n (xnk )k
a ∈ S a �= 0
lim
k→∞ � xnk − a �2= 0
� a �∗ ≤ � a − xnk �∗ + � xnk �∗
≤ C � a − xnk �2 + � xnk �∗
� a �∗ ≤ lim
k→∞ C � a − xnk �2 + lim � xnk �∗
k→∞
= 0
a = 0
c>0
a �= 0

x �= 0
x =0
K n
x
∈ S
� x �2
� �
�
c ≤ �
x �
�
��
x �2
�
∗
c � x �2 ≤ � x �∗
c � x �2≤� x �∗≤ C � x �2
K f : K n → V
K n
� x �f := � f(x) �
� x � ∗ f := � f(x) � ∗
� x �f = � f(x) �≥ 0
� cx �f = � f(cx) � = � cf(x) �
= |c| �f(x) �= |c| �x �f
� x + y �f = � f(x + y) �=� f(x)+f(y) �
≤ � f(x) � + � f(y) �=� x �f + � y �f
� x �f =0 ⇐⇒ � f(x) �= 0 ⇐⇒ f(x) =0
c, C > 0
⇐⇒ x =0
∀x ∈ K n : c � x �f ≤� x � ∗ f ≤ C � x �f
∀x ∈ K n : c � f(x) �≤� f(x) � ∗ ≤ C � f(x) �
∀y ∈ V : c � y �≤� y � ∗ ≤ C � y �

m × n
f : R n → R m
f(a + s) =f(a)+As
f : R n → R m
U ⊂ R n
f : U ⊂ R n → R m a ∈ U ⇐⇒
∃ A ∈ M(m × n)
�
�
�
�
s �
� (A − B) �
� s ��
� f(a + s) − f(a) − As �
lim
=0
�s�→0,s�=0 � s �
ε>0
Df(a) :=A
� As − Bs �
=
� s �
= � f(a + s) − f(a) − Bs − (f(a + s) − f(a) − As) �
� s �
� f(a + s) − f(a) − Bs �
≤
� s �
< ε ε
+ = ε
2 2
∀s �= 0:
+ � f(a + s) − f(a) − As �
�
�
�
�
s �
� (A − B) �
� s ��
= 0
� A − B � = 0
A = B
U ⊂ R n
f : U ⊂ R n → R m a ∈ U ⇐⇒
∃ m × n ∃r : U → R m
lim
x→a,x�=a
f(x) = f(a)+A(x − a)+r(x)
r(a) = 0
� r(x) �
� x − a �
= 0
� s �

⇐
⇒
lim
x→a,x�=a
N : xn �= a
r(x) =
� r(x) �
� x − a �
A = Df(a)
� f(x) − f(a) − Df(a)(x − a) x �= a
0 x = a
� f(x) − f(a) − Df(a)(x − a) �
= lim
x→a,x�=a
� x − a �
= lim
�s�→0,s�=0
= 0
� f(a + s) − f(a) − As �
� s �
� f(a + s) − f(a) − As �
lim
�s�→0,s�=0 � s �
� f(x) − f(a) − A(x − a) �
= lim
x→a,x�=a � x − a �
� r(x) �
= lim
x→a,x�=a � x − a � =0
A = Df(a)
(xn)n U ⊂ R n limn→∞ xn = a ∀n ∈
lim
n→∞
� r(xn) �
� xn − a � =0
⇐⇒ ∃ (cn)n R limn→∞ cn =0
⇒
� r(xn) �= cn � xn − a �
cn := � r(xn) �
� xn − a �
lim
n→∞ cn
� r(xn) �
= lim
n→∞ � xn − a � =0

⇐
a ∈ U
lim
n→∞
� r(xn) �
= lim
� xn − a � n→∞ cn =0
U ⊂ R n f : U ⊂ R n → R m
limn→∞ xn = a xn ∈ U ∀n ∈ N : xn �= a
0 ≤ lim
n→∞ � f(xn) − f(a) �
= lim
n→∞ � Df(a)(xn − a)+r(xn) �
≤ � Df(a) � lim
n→∞ � xn − a � + lim
n→∞
� �� �
=0
� r(xn) �
� �� �
=cn�xn−a�
= lim
n→∞ cn lim
n→∞
� �� �
=0
� xn − a �
� �� �
=0
= 0
lim
n→∞ f(xn) =f(a)
U ⊂ R n ,V ⊂ R m f : U → V,g : V → R k
x f(x) g ◦ f
D(g ◦ f)(x) =Dg(f(x)) · Df(x)
limn→∞ xn = x xn �= x f(x) �= f(xn)
g(f(xn)) Def
= g(f(x)) + Dg(f(x)) · (f(xn) − f(x)) + r1(f(xn))
Def
= g(f(x)) + Dg(f(x)) · [Df(x)(xn − x)+r2(xn)] + r1(f(xn))
= g(f(x)) + Dg(f(x)) · Df(x)(xn − x)+r3(xn)
(cn)n
R
r3(xn) = Dg(f(x))r2(xn)+r1(f(xn))
� r1(f(xn)) � = cn � f(xn) − f(x) �
lim
n→∞ cn = 0

� r3(xn) �
0 ≤ lim
n→∞ � xn − x �
g ◦ f
= lim
n→∞
� Dg(f(x))r2(xn)+r1(f(xn)) �
� xn − x �
≤ � Dg(f(x)) � lim
n→∞
r2(xn)
xn − x
� �� �
=0
= lim
n→∞ cn
� f(xn) − f(x) �
� xn − x �
= lim
n→∞ cn
� Df(x)(xn − x)+r2(xn) �
� xn − x �
⎛
� r1(f(xn)) �
+ lim
n→∞ � xn − x �
= lim
n→∞ cn
⎜
� �� �
⎝
=0
lim
� Df(x) �� xn − x � � r2(xn) �
⎟
+ lim
⎟
n→∞ � xn − x � n→∞ � xn − x �⎠
� �� � � �� �
=�Df(x)�
=0
= 0
D(g ◦ f)(x) =Dg(f(x)) · Df(x)
a ∈ U
�
� f(a + s) − f(a) − As �
∀�s �< δ,s�= 0⇒
⇐⇒ ∀ε>0 ∃δ >0:
� s �
�
⇐⇒ lim
n→∞ sn
� f(a + sn) − f(a) − Asn �
=0,sn �= 0⇒ lim
n→∞ � sn �
f : U ⊂ R n → R m x ∈ U ⇐⇒
∀1 ≤ i ≤ m : fi : U ⊂ R n → R m x ∈ U
A := Df(x)
fi(x + s) =fi(x)+
n�
aijsj + ri(s)
j=1
ri(s)
lim
s→0 � s � =0
⎞
�

limk→∞ sk =0
R n , R m
lim g(s) =0
s→0
� f(x + sk) − f(x) − Ask �
lim
=0
k→∞ � sk �
� f(x + sk) − f(x) − Ask �∞
⇐⇒ lim
k→∞
⇐⇒ ∀1 ≤ i ≤ n : lim
k→∞
� sk �∞
=0
|fi(x + sk) − fi(x) − (Ask)i|
� sk �
=0
⇐⇒
|fi(x + sk) − fi(x) −
∀1 ≤ i ≤ n : lim
k→∞
�n j=1 aijsk,j|
=0
� sk �
⇐⇒
ri(sk)
∀1 ≤ i ≤ n : lim
k→∞ � sk � =0
⇐⇒
ri(s)
∀1 ≤ i ≤ n :lim
s→0 � s � =0

∀x ∈ U
R 2 \{0}
U ⊂ R n
f : U ⊂ R n → R, (x1,...,xn) ↦→ f(x1,...,xn)
i =1,...,n
∂f
∂xi
x ∈ U ⇐⇒
f(x + hei) − f(x)
lim
h→0 h
f(x + hei) − f(x)
(x) := lim
h→0 h
⇐⇒
∂f
= Dif : U ⊂ R
∂xi
n → R
∂ n f
∂x1 ...∂xn
F : R 2 ⎧
⎨
→ R, (x1,x2) ↦→
⎩
∂F
∂x1
∂F
=
∂x1
= ∂
...
∂x1
∂
f
∂xn
xj
⇐⇒ ∀1 ≤ i ≤ n
x1x2
(x 2 1 + x2 2 )2 x �= 0
0 x =0
x2
(x2 1 + x2 x1x22x1
− 2
2 )2 (x2 1 + x22 )3
F (hei) − F (0) 0 − 0
(0) = lim
= lim
h→0 h
h→0 h =0

x, y ∈ B(0, 2δ)
F : R 2 → R
ak =
lim
k→∞ ak = 0
� �
1 1
,
k k
lim
k→∞ F (ak) = lim
k→∞
= lim
k→∞
(ak)k∈N
1 1
k k
� 1
k 2 + 1
k 2
k 2
4
= ∞
U ⊂ R n f : U → R
a ∈ U 1 ≤ i, j ≤ n
∂ ∂
f(a) =
∂xj ∂xi
∂ ∂
f(a)
∂xi ∂xj
� 2
δ>0 B(a, 2δ) ⊂ U
(x + ai,y+ aj) =(a1,...,x+ ai,...,y+
aj
� �� � � �� � ,...,an)
(ai + s, aj + y) ∈ U
=
∂2f (ai + s, aj + y)
∂xj∂xi
∃t |t| < |y|
∂
f(ai + s, aj + y) −
∂xi
∂
f(ai + s, aj)
∂xi
� aj+y
∂2f (ai + s, xj)dxj
∂xj∂xi
aj
∃t
= (aj + y − aj) · ∂2f (ai + s, aj + t)
∂xj∂xi
= y · ∂2f (ai + s, aj + t)
∂xj∂xi

∂f
(ai + x, aj + y) −
∂xi
∂f
(ai + x, aj)
∂xi
∃s |s| < |x|
f(ai + x, aj + y) − f(ai + x, aj) − f(ai,aj + y)+f(ai,aj)
� ai+x �
∂f(xi,aj + y)
=
−
ai ∂xi
∂f(xi,aj)
�
dxi
∂xi
�
∂f
= (ai + x − ai) · (ai + s, aj + y) −
∂xi
∂f
�
(ai + s, aj)
∂xi
= xy · ∂2f (ai + s, aj + t)
∂xj∂xi
(ai + x, aj + t ′ ) ∈ U
∂ 2 f
∂xi∂xj
∃s ′ |s ′ | < |x|
∂f
∂xj
(ai + x, aj + t ′ )
(ai + x, aj + t ′ ) − ∂f
� ai+x
∂2f(xi,aj + t ′ )
=
ai ∂xi∂xj
= (ai + x − ai) · ∂2f ∂xi∂xj
(ai,aj + t
∂xj
′ )
dxi
= x · ∂2f (ai + s
∂xi∂xj
′ ,aj + t ′ )
(ai + s ′ ,aj + t ′ )
∂f
(ai + x, aj + y) −
∂xj
∂f
(ai,aj + y)
∂xj
∃t ′ |t ′ | < |y|
f(ai + x, aj + y) − f(ai + x, aj) − f(ai,aj + y)+f(ai,aj)
� aj+y �
∂f(ai + x, xj)
=
−
aj ∂xj
∂f(ai,xj)
�
dxj
∂xj
� �
∂f
= (aj + y − aj) ·
= xy · ∂2f (ai + s
∂xi∂xj
′ ,aj + t ′ )
(ai + x, aj + t
∂xj
′ ) − ∂f
(ai,aj + t
∂xi
′ )

xy �= 0
∂2f (ai + s, aj + t) =
∂xj∂xi
∂2f ∂xi∂xj
f(ai + s ′ ,aj + t ′ )
∂2f (ai,aj)
∂xj∂xi
=
∂
lim
(x,y)→0
2f (ai + s, aj + t)
∂xj∂xi
=
∂
lim
(x,y)→0
2f ∂xi∂xj
=
∂2f (ai,aj)
∂xi∂xj
(ai + s ′ ,aj + t ′ )
f : U ⊂ R n → R m x ∈ U
fi : U → R 1 ≤ i ≤ m
∂fi
∂xj
fi(x + s) =fi(x)+
(x) =aij
fi
n�
aijsj + ri(s)
j=1
fi(x + hej) =fi(x)+haij + ri(hej)
∂fi
∂xj
(x) =
fi(x + hej) − fi(x)
lim
h→0 h
=
ri(hej)
aij +lim
h→0 h
= aij
U ⊂ R n f : U → R

(s1,...,sn) ∈ R n � s �< δ
ti ∈ [0, 1]
z (0)
∀1 ≤ i ≤ n : z (i)
δ>0 B(x, δ) ⊂ U
:= x
:= x +
i�
k=1
skek
∀1 ≤ i ≤ n : z (i) − z (i−1) = siei
si �= 0: f(z(i) ) − f(z (i−1) )
si
= Dif(z (i−1) + tisiei)
∀si ∈ B(x, δ) : f(z (i) ) − f(z (i−1) ) = si · Dif(z (i−1) + tisiei)
f(x + s) − f(x) =
Dif
=
=
n�
i=1
n�
i=1
n�
i=1
0 ≤ lim
s→0
n�
≤ lim
Dif = 0
(f(z (i) ) − f(z (i−1) ))
Dif(z (i−1) + tisiei) · si
aisi +
�n i=1
n�
i=1
(Dif(z (i−1) + tisiei) − Dif(x)) · si
�
�Dif(z (i−1) + tisiei) − Dif(x) � � ·|si|
� s �
s→0
i=1
|Dif(z (i−1) + tisiei) − Dif(x)|
U ⊂ R n f : U → R
f : U → R

v :[a, b] ⊂ R → Rn �
��
�
b �
� �
� v(t)dt�
≤
�
a
u := � b
a v(t)dt
� 2
� u � 2 2 = 〈u, u〉 =
=
≤
� b
a i=1
� b
a
= � u �2
� b
a
� v(t) �2 dt
n�
�� �
b
vi(t)dt
i=1
a
n�
vi(t)uidt =
� b
� v(t) �2� u �2 dt
� b
a
a
� v(t) �2 dt
M := sup � Df(x + ts) �2
t∈[0,1]
� f(x + s) − f(x) �2≤ M � s �2
� f(x + s) − f(x) �2 =
≤
≤
�
�
�
�
� 1
0
� 1
0
� 1
0
≤ M � s �2
ui
〈v(t),u〉 dt
�
�
Df(x + ts)sdt�
� 2
� Df(x + ts)s �2 dt
� Df(x + ts) �2� s �2 dt

R 2 C
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
R 2
R 2
+:R 2 × R 2 → R 2 ��
x1
,
y1
· : R 2 × R 2 → R 2 ��
x1
,
y1
� x1
�
,
�
,
� x2
� x2
y2
y2
C
C
R
�� � �
x1 + x2
↦→
y1 + y2
�� � �
x1x2 − y1y2
↦→
� � 0
0
��
x1
� �� � � ��
x2 x3
+ + =
y1 y2 y3
� � � � � � � �
x1 x2 x2 x1
+ = +
y1 y2 y2 y1
� � � � � �
x 0 x
+ =
y 0 y
� � � � � �
x −x 0
+ =
y −y 0
� x1
y1
x1y2 + x2y1
�
+
� x2
y2
� � 1
0
�� �
x1 x2
��
+
� �� � � �� � ��
x2 x3
· · = · ·
y1 y2 y3 y1 y2
� � � � � � � �
x1 x2 x2 x1
· = ·
y1 y2 y2 y1
� � � � � �
x 1 x
· =
y 0 y
⎛ ⎞
� �
� � � � � �
x ⎜ ⎟ 1 x 0
· ⎝ ⎠ =
�=
y
0 y 0
x
x 2 + y 2
−y
x 2 + y 2
� x3
� x3
y3
y3
�
�

9.)
� x1
y1
�
·
=
=
=
=
�� x2
y2
� ��
x1 x2
� �
x
y
y1
�
+
R 2
� x3
y3
��
=
� x1
y1
�
·
� x2
y2
�
+
� x1
y1
� � �� � � ��
x1 x2 x3
· ,
y1 y2 y3
� � � �
x1 x2x3 − y2y3
·
y1 x2y3 + y2x3
� �
x1x2x3 − x1y2y3 − y1x2y3 − y1y2x3
x1x2y3 + x1y2x3 + y1x2x3 − y1y2y3
� �
x1x2x3 − y1y2x3 − x1y2y3 − y1x2y3
x1x2y3 − y1y2y3 + x1y2x3 + y1x2x3
� � � �
x1x2 − y1y2 x3
·
x1y2 + y1x2
y2
y3
� � �
x1x2 − y1y2
=
=
� �� �
x 1
y 0
⎛
⎜
· ⎝
x
x 2 + y 2
−y
x 2 + y 2
=
x1y2 + x2y1
⎞
� �
x · 1 − y · 0
=
x · 0+y · 1
⎟
⎠ =
⎛
⎜
⎝
� ��
x2 x1
y2
x 2 + y 2
x 2 + y 2
x(−y)+yx
x 2 + y 2
� �
x
y
⎞
y1
⎟
⎠ =
�
�
·
� �
1
0
� x3
y3
�

=
=
=
=
� ���
x1 x2
y1
�
x1
y1
�
+
y2
�� �
x2 + x3
y2 + y3
� x3
y3
��
� �
x1x2 + x1x3 − y1y2 − y1y3
x1y2 + x1y3 + y1x2 + y1x3
� � � �
x1x2 − y1y2 x1x3 − y1y3
+
x1y2 + y1x2 x1y3 − y1x3
� �� � � �� �
x1 x2 x1 x3
+
y1
y2
C R
� � � �
x1 x2
± =
0 0
� � � �
x1 x2
· =
0 0
� �−1 x1
=
0
� �
x1
·
0
� �
x2
0
� �−1 x1
0
=
=
y1
y3
� �
x1 ± x2
0
� �
x1 · x2
0
� −1�
x1 0
� � � �
x1x2 +0· 0 x1 · x2
=
x1 · 0+x2 · 0 0
⎛ x1
⎞
� −1�
⎜ ⎟ x1 ⎝ ⎠ =
0
x 2 1 +02
0
x 2 1 +02
1 :=
∀x ∈ R : x :=
i :=
� �
1
0
� �
x
0
� �
0
1

x + iy =
� �
x
y
x + iy = x · 1+y · i
� � � � � �� �
x 1 y 0
= · +
0 0 0 1
� � � �
x 0 · y − 1 · 0
= +
0 0 · 0+1· y
� �
x
=
y
i 2 = −1
i 2 =
� �� �
0 0
1 1
z = x + iy ∈ C
=
z := x − iy
z = x + iy
z = z ⇐⇒ z = x
x, y ∈ R
x = 1
2 (z + z)
z = z
1 y = 2i (z − z)
z1 + z2 = z1 + z2
z1z2 = z1z2
� �
−1
= −1
0
z = z ⇐⇒ x + iy = x − iy ⇐⇒ 2iy =0
i�=0
⇐⇒ y =0 ⇐⇒ z = x
1
(z + z)
2
=
1
(x + iy + x − iy) =1 2x = x
2 2
1
(z − z)
2i
=
1
1
(x + iy − (x − iy)) = 2iy = y
2i 2i

z = x − iy = x + i(−y)
= x − i(−y) =x + iy = z
z1 + z2 = x1 + iy1 + x2 + iy2
= (x1 + x2)+i(y1 + y2)
= x1 + x2 − i(y1 + y2)
= x1 − iy1 + x2 − iy2
= z1 + z2
z1z2 = (x1 + iy1)(x2 + iy2)
= x1x2 − y1y2 + i(x1y2 + x2y1)
= x1x2 − y1y2 − i(x1y2 + x2y1)
= (x1−iy1)(x2 − iy2)
= z1 · z2
C R2 ��
��
�
|z| := �
x ���2
� y
z = x + iy ∈ C
|·|: C → [0, ∞),z ↦→ √ zz = � x 2 + y 2
zz =(x + iy)(x − iy) =x 2 + y 2 ≥ 0
≥ 0
z = x + iy
z ∈ R |z|C = |z|R
|z| = |z|
x ≤|z| y ≤|z|
|z1z2| = |z1||z2|
|z| =0 ⇐⇒ z =0
|z1 + z2| ≤|z1| + |z2|

|z| = √ zz ≥ 0
|z| = √ zz = � (x + i0)(x − i0) = √ x 2 = |x|
|z| = √ zz =
�
z · z = |z|
x 2 ≤ x 2 + y 2
y 2 ≤ x 2 + y 2
x ≤ |x| = √ x2 ≤ � x2 + y2 = |z|
y ≤ |y| = � y2 ≤ � x2 + y2 = |z|
|z1z2| 2 = z1z2z1z2
= z1z2z1z2
= z1z1z2z2
= |z1| 2 |z2| 2
|z| =0 ⇐⇒ � x2 + y2 =0 ⇐⇒ x 2 + y 2 =0
⇐⇒ x =0 y =0 ⇐⇒ z =0
z1z2 = x + iy
1
2 (z1z2 + z2z1) =x ≤|z1z2| = |z1|·|z2|
|z1 + z2| 2 = (z1 + z2)(z1 + z2)
= z1z1 + z1z2 + z2z1 + z2z2
≤ |z1| 2 +2|z1||z2| + |z2| 2
= (|z1| + |z2|) 2
|z1 + z2| ≤|z1| + |z2|

C
(cn)n C c ⇐⇒
∀ε >0 ∃n0 ∀n ≥ n0 : |cn − c| 0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : |cn − cm|

R
C (cn)n
cn = an + ibn
(an)n (bn)n R
a b R
(cn)n
a + ib
(cn)n
C
∃K >0 ∀n ∈ N : |cn| ≤K
cn = an + ibn (an)n (bn)n
R
|cn| = � a 2 n + b 2 n
lim
n→∞ |cn| =
=
lim
�
�
� a 2 n + b 2 n
n→∞
lim
n→∞ an
�2 = � a 2 + b 2
(|cn|)n
(cn)n (dn)n C
�
+ lim
n→∞ bn
�2 lim
n→∞ (cn + dn) = lim
n→∞ cn + lim
n→∞ dn
lim
n→∞ (cndn) =
d �= 0 dn �= 0 n ≥ n0
n ≥ max(n1,n2)
lim
n→∞
1
dn
�
lim
n→∞ cn
=
��
1
limn→∞ dn
lim
n→∞ dn
limn→∞ cn = c limn→∞ dn = d n1,n2 ∈ N
∀n ≥ n1 : |cn − c| < ε
2
∀n ≥ n2 : |dn − d| < ε
2
|cn + dn − (c + d)| ≤|cn− c| + |dn − d| < ε ε
+ = ε
2 2
�

K1
(|cn|)n
K := max(K1, |d|)
limn→∞ cn = c limn→∞ dn = d n1,n2 ∈ N
n ≥ max(n1,n2)
∀n ≥ n1 : |cn − c| < ε
2K
∀n ≥ n2 : |dn − d| < ε
2K
|cndn − cd| ≤ |cndn−cnd + cnd − cd|
≤
<
|cn||dn − d| + |d||cn − c|
K ε ε
+ K = ε
2K 2K
(|dn|)n
|d|
∃n0 ∈ N ∀n ≥ n0 : |dn| ≥ |d|
2
limn→∞ dn = d n0 ∈ N
∀n ≥ n0 : |dn − d| < ε|d|2
2
�
�
�
1
� − 1
�
�
�
d�
=
dn
1
|d − dn|
|dnd|
2
<
|d| 2
ε|d| 2
2
= ε

� ∞
k=0 ck
⇐⇒
C
R
(c0,c1,...) C
�
n�
(c0,c0 + c1,c0 + c1 + c2,...)n =
� ∞
k=0 ck
k=0
(ck)k
C �∞ k=0 ck
�
� n�
�
∀ε >0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : �
�
� n�
k=0
� n�
k=0
ck
ck
�
�
n
n
k=m
�
� n�
�
⇐⇒ ∀ε>0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : �
�
k=0
�
� n�
�
⇐⇒ ∀ε>0 ∃n0 ∈ N ∀n ≥ m ≥ n0 : �
�
� ∞
n=0 cn
m = n − 1 m, n > n0
�
� n� �
|cn| = �
�
�
� ∞
k=0 |ck|
k=0
n−1
ck − ck
k=0
k=m
�
�
�
�
�

�
� n�
�
�
� �
∀ε >0 ∃n0 ∀n ≥ m ≥ n0 : � |ck| �
� �

�∞ n=0 an an ≥ 0 (cn)n
C |cn| ≤an
� ∞
n=0 |cn|
�
� n�
�
∀ε >0 ∃n0 ∀n ≥ m ≥ n0 : �
�
∀ε >0 ∃n0 ∀n ≥ m ≥ n0
�
� n�
�
�
� �
� ck�
� � ≤
n → n +1
n =0
k=m
z ∈ C\{0}
�
n+1
z k
k=0
n�
�
� n� �
|ck| ≤�
�
k=m
n�
z k =
k=0
=
=
=
=
z ∈ C |z| < 1
z 0 =1=
k=m
1 − zn+1
1 − z
1 − z
1 − z
n�
z k + z n+1
k=0
|z| ∈R 0 ≤|z| < 1
k=m
ak
ak
�
�
�
�
�

lim
n�
n→∞
k=0
|z| < 1
� ∞
n=0 |cn|
∞�
k=0
z k = 1
1 − z
z k 1 − z
= lim
n→∞
n+1
1 − z = 1 − limn→∞ zn+1 =
1 − z
1
1 − z
0

∞�
n=0
cn =
�∞ n=0 |an|, �∞ n=0 |bn|
�∞ n=0 bn
Dn :=
|D|n :=
Qn :=
|Q|n :=
� ∞�
n=0
n�
k=0
an
ck =
n�
|ck|
k=0
� n�
k=0
ak
�� ∞�
n�
n=0
k�
k=0 j=0
�� n�
k=0
bn
�
ak−jbj
bk
�
�
n�
��
n�
�
|ak| |bk|
k=0
k=0
lim
n→∞ |Q|n
�
n�
��
n�
�
= lim |ak| |bk|
n→∞
k=0 k=0
�
n�
� �
n�
�
= lim |ak| lim |bk|
n→∞
n→∞
k=0
k=0
�
∞�
��
∞�
�
= |ak| |bk| < ∞
(|Q|n)n
∀ε >0 ∃n0 ∀n ≥ n0
k=0
k=0
�
||Q|n −|Q|n0 | = |ai|·|bj|−
i,j≤n
�
|ai|·|bj| <
i,j≤n0
ε
2
� ∞
n=0 an

n>2n0
n>2n0
||Q|n −|D|n| = �
∞�
n=0
=
≤
0≤i,j≤n
�
0≤i,j≤n,i+j>n
�
n0≤i,j≤n
|ai| |bj|− �
|ai|·|bj|
i+j≤n
|ai|·|bj|
≤ ||Q|n −|Q|n0| < ε
2
|cn| = lim
n→∞ |D|n = lim
n→∞ |Q|n < ∞
|Qn − Dn| =
=
=
≤
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
0≤i,j≤n
�
0≤i,j≤n
�
aibj −
n�
k�
k=0 j=0
aibj − �
0≤i,j≤n,i+j>n
�
n0≤i,j≤n
i+j≤n
aibj
|ai|·|bj|
ε
= ||Qn|−|Qn0 || <
2
lim
n→∞ Qn = lim
n→∞ Dn
�
�
�
�
�
�
|ai||bj|
ak−jbj
aibj
�
�
�
�
�
�
�
�
�
�
�
�

∞�
k=0
ck
Def
= lim
n→∞ Dn
= lim
n→∞ Qn
�
n�
= lim
n→∞
k=0
�
n�
= lim
n→∞
k=0
�
∞�
=
n=0
an
ak
ak
�� n�
�
�� ∞�
n=0
k=0
lim
n→∞
bn
�
bk
�
� n�
k=0
bk
�

n → n +1
(x + y) n+1
n =1
x, y ∈ C
n�
k=0
C
� �
n
x
k
n−k y k = 1
(x + y)n
n!
x + y = x 1−0 · y 0
����
=1
+ x 0
����
=1
·y 1−0
= (x + y) n (x + y)
�
n� � �
n
=
x
k
k=0
n−k y k
�
(x + y)
n�
� �
n
=
x
k
k=0
n+1−k y k n�
� �
n
+ x
k
k=0
n−k y k+1
= x n+1 n�
� �
n
+ x
k
k=1
n+1−k y k n−1 �
� �
n
+ x
k
k=0
n−k y k+1 + y n+1
= x n+1 n�
� �
n
+ x
k
k=1
n+1−k y k n�
� �
n
+
x
k − 1
k=1
n−(k−1) y k + y n+1
� �
n +1
=
x
0
n+1 n�
� �
n +1
+
x
k
k=1
n+1−k y k � �
n +1
+ y
n +1
n+1
n+1 �
� �
n +1
=
x
k
n+1−k y k
k=0
z ∈ C
e z := exp(z)
exp : C → C,z ↦→
e := exp(1) =
�∞ |z|
n=0
n
n!
∞�
n=0
∞�
n=0
1
n!
z n
n!

� ∞
n=0
n ≥ 2|z|
� �
�an+1
�
� �
� � = |z|n+1n! |z| n (n +1)!
an
|z| n
n!
n=0 zn
n!
� ∞
�∞ |z1|
n=0
n
n!
cn :=
n�
k=0
z1,z2 ∈ C
|z| 1
= ≤
n +1 2
exp(z1)exp(z2) =exp(z1 + z2)
�∞ |z2|
n=0
n
n!
z n−k
1 zk 2 1
=
(n − k)!k! n!
exp(z1 + z2) =
z ∈ C
=
n�
k=0
� �
n
k
z n−k
1 z k 2 = 1
n! (z1 + z2) n
∞� (z1 + z2) n
n=0
� ∞�
n=0
z n 1
n!
n!
��
∞�
n=0
= exp(z1)exp(z2)
exp(z) �= 0
exp(−z) =
1
exp(z)
z n 2
n!
exp(z)exp(−z) =exp(z − z) =exp(0)=1
z ∈ C exp(z) =exp(z)
�

|rN+1(z)| =
n� zk k! =
n� zk k!
k=0
exp(z) = lim
∀|z| ≤
=
N+2≤N+k+1
≤
|z| 1
N+2 ≤ 2
≤
=
exp
limn→∞ zn =0
= lim
exp(z) =
N +2
2
k=0
n�
n→∞
k=0
n�
n→∞
k=0
N�
n=0
z k
k!
z k
k!
= lim
n�
n→∞
k=0
= exp(z)
zn + rN+1(z)
n!
: |rN+1(z)| ≤2 |z|N+1
(N +1)!
z k
k!
∞� |z|
n=N+1
n
n!
|z| N+1
�
∞�
|z|
1+
(N +1)!
k=1
k
�
(N +2)· ...· (N + k +1)
|z| N+1 ∞� |z|
(N +1)!
k
(N +2) k
|z| N+1
(N +1)!
|z| N+1
(N +1)!
exp : C → C,z ↦→ exp(z)
k=0
∞�
k=0
1
1
2 k
1 − 1
2
=2 |z|N+1
(N +1)!
∃n0 ∀n ≥ n0 : |zn| < 1

exp
∀|zn| ≤ 0+2
: |r1(zn)| ≤
2
2|zn| 1
0 ≤ lim
n→∞ | exp(zn) − 1|
= |r1(zn)|
|zn|≤1
≤ lim
n→∞ 2|zn| =0
lim
n→∞ exp zn =1
limn→∞ zn = p limn→∞(zn − p) =0
lim
n→∞ exp(zn) = lim
n→∞ exp(zn − p + p)
1!
= exp(p) lim
n→∞ exp(zn − p)
� �� �
=1
= exp(p)

cos : R → R,x↦→ exp(ix)+exp(−ix)
2
exp(ix) − exp(−ix)
sin : R → R,x↦→
2i
exp(ix) =cosx + i sin x
cos x + i sin x = exp(ix)+exp(−ix)
2
= exp(ix)
exp
exp(ix) − exp(−ix)
+ i
2i
cos(R) ⊂ R sin(R) ⊂ R
cos : R → R sin : R → R R
lim
n→∞ exp(ixn) =exp(ilim n→∞ xn) =exp(ix)
lim
n→∞ cos xn =
1
lim
n→∞ 2 (exp(ixn)+exp(−ixn))
= 1
2 (exp(ix)+exp(−ix))
= cosx
lim
n→∞ sin xn =
1
lim
n→∞ 2i (exp(ixn) − exp(−ixn))
= 1
(exp(ix) − exp(−ix))
2i
= sinx
∀x ∈ R
a) cos(−x) =cosx
b) sin(−x) =− sin x
c) | exp(ix)| =1
d) cos 2 x +sin 2 x =1

cos(−x) =
exp(−i(−x)) + exp(i(−x))
2
= exp(ix)+exp(−ix)
= cos(x)
2
sin(−x) =
exp(−i(−x)) − exp(i(−x))
2i
=
exp(ix) − exp(−ix)
− = − sin(x)
2i
| exp(ix)| 2
cos 2 x +sin 2 x
Def
= exp(ix)exp(−ix)
= exp(ix +(−ix)) = exp(0) = 1
� �2 1
=(−i)
i
2 = i 2 = −1
� �2 �
exp(ix)+exp(−ix) exp(ix) − exp(−ix)
=
+
2
2i
= 1 � 2 2
(exp(ix)) +2exp(ix)exp(−ix)+(exp(−ix))
4
−(exp(ix)) 2 − 2exp(ix)exp(−ix)+(exp(−ix)) 2�
= exp(ix)exp(−ix)
= exp(ix − ix)
= exp(0) = 1
x, y ∈ R
cos(x + y) = cosxcos y − sin x sin y
sin(x + y) = sinxcos y +cosxsin y
sin x − sin y
cos x − cos y
=
=
x + y x − y
2cos sin
2 2
x + y x − y
−2sin sin
2 2
� 2

cos(x + y)+i sin(x + y)
= exp(i(x + y)) = exp(ix)exp(iy)
= (cosx + i sin x)(cos y + i sin y)
= (cosxcos y − sin x sin y)+i(sin x cos y +cosxsin y)
u := x+y
2 v := x−y
2
x = u + v y = u − v
sin x − sin y
= sin(u + v) − sin(u − v)
= (sinucos v +cosusin v) − (sin u cos(−v)+cosusin(−v)) = 2cosusin v
x + y x − y
= 2cos sin
2 2
cos x − cos y
= cos(u + v) − cos(u − v)
= cosucos v − sin u sin v − (cos u cos(−v) − sin u sin(−v) )
� �� � � �� �
=cos v
=− sin v
= −2sinusin v
x + y x − y
= −2sin sin
2 2
x ∈ R
∞�
k=0
|x| 2k
(2k)!
cos x =
sin x =
∞�
k=0
|x| 2k+1
(2k +1)!
∞�
k x2k
(−1)
(2k)!
k=0
∞�
k x2k+1
(−1)
(2k +1)!
k=0

� ∞
�∞ |x|
k=0
k
k!
x2k+1
k=0 (−1)k (2k+1)!
�∞ |x|
k=0
k
k!
r2n+2(x) =
=
cos x + i sin x
exp(ix)
=
∞� (ix)
n=0
n
n! =
∞�
n xn
i
n!
n=0
=
∞�
∞�
=
k=0
k=0
2k x2k
i
(2k)! +
k=0
2k+1 x2k+1
i
(2k +1)!
∞�
∞�
k x2k
k x2k+1
(−1) + i (−1)
(2k)! (2k +1)!
cos x =
sin x =
k=0
k=0
n�
k x2k
(−1)
(2k)!
n�
k x2k+1
(−1)
(2k +1)!
k=0
+ r2n+2(x)
+ r2n+3(x)
∀|x| ≤2n +3: |r2n+2(x)| ≤ |x|2n+2
(2n +2)!
∀|x| ≤2n +4: |r2n+3(x)| ≤ |x|2n+3
(2n +3)!
∞�
k=n+1
k x2k
(−1)
∞�
k=n+1
n+1 x2n+2
= (−1)
(2n +2)!
k x2k
(−1)
(2k)!
�
∞�
1+
k=1
(2k)!
�∞ x2k
k=0 (−1)k (2k)!
x2k (−1) k
�
(2n +3)· ...· (2n +2(k +1))

k ≥ 1
a0 = 1
ak =
x2k ∈ [0, ∞)
(2n + 3)(2n +4)...(2n +2(k +1))
|x| ≤2n +3
r2n+3(x)
n+1 x2n+2
r2n+2(x) =(−1)
(2n +2)!
ak = ak−1
∞�
k=0
(−1) k ak
x2 (2n +2k + 1)(2n +2k +2)
1=a0 >a1 >a2 >...>0
∞�
k=0
0 ≤ a0 − a1 ≤
(−1) k ak
∞�
(−1) k ak ≤ a0 =1
k=0
|r2n+2(x)| ≤ |x|2n+2
(2n +2)!
∞�
k=n+1
∞�
k x2k+1
= (−1)
(2k +1)!
k=n+1
�
∞�
n+1 x2n+3
= (−1) 1+
(2n +3)!
k x2k+1
(−1)
k=1
(2k +1)!
x2k (−1) k
�
(2n +4)· ...· (2n +2(k +1)+1)

k ≥ 1
a0 = 1
ak =
x2k ∈ [0, ∞)
(2n + 4)(2n +5)...(2n +2(k +1)+1)
|x| ≤2n +4
n+1 x2n+3
r2n+3(x) =(−1)
(2n +3)!
ak = ak−1
∞�
k=0
(−1) k ak
x2 (2n +2k + 2)(2n +2k +3)
1=a0 >a1 >a2 >...>0
∞�
k=0
0 ≤ a0 − a1 =
sin cos R
(−1) k ak
∞�
(−1) k ak ≤ a0 =1
k=0
|r2n+3(x)| ≤ |x|2n+3
(2n +3)!
a) sin ′ (x) =cos(x)
b) cos ′ (x) =sin(x)
sin x = x + r3(x)
∀|x| < 2n +4: |r2n+3(x)| ≤ |x|2n+3
(2n +3)!
∀|x| < 4: |r3(x)| ≤ |x|3
3!

0 < ∀|h| < 1
� �
�
�
sin h �
� − 1�
h � =
� �
�
�
sin h − h�
�
|r3(h)|
� h � =
|h|
≤ |h|3 |h|2
=
3!|h| 6
cos
cos 2 ≤− 1
3
∀x ∈ (0, 2] : sin x>0
cos [0, 2]
� �
�
0 ≤ lim �
sin h − h�
�
h→0 � h �
sin h
lim
h→0 h =1
|h|
≤ lim
h→0
2
6 =0
sin ′ sin(x + h) − sin(x)
(x) := lim
h→0
= lim
h→0
= lim
h→0 cos
= cosx
h
2cos 2x+h
2
h
�
x + h
2
sin h
2
�
lim
h→0
cos ′ cos(x + h) − cos(x)
(x) := lim
h→0
= lim
h→0
= − lim
h→0 sin
= − sin x
h
−2sin 2x+h
2
h
�
x + h
2
sin h
2
�
lim
h→0
sin h
2
h
2
sin h
2
h
2

cos x = 1− x2
+ r4(x)
2
∀|x| < 2n +3: |r2n+2(x)| ≤ |x|2n+2
(2n +2)!
∀|x| < 5: |r4(x)| ≤ |x|4
4!
cos 2 = 1 − 2+r4(2)
|r4(2)| ≤ 24 2
=
24 3
− 2
3 ≤ r4(2) ≤ 2
3
1 − 2 − 2
3 ≤ 1 − 2+r4(2) ≤ 1 − 2+ 2
3
− 5
3
≤ cos 2 ≤−1
3
sin x = x + r3(x)
∀|x| < 2n +4: |r2n+3(x)| ≤ |x|2n+3
(2n +3)!
∀|x| < 4: |r3(x)| ≤ |x|3
3!
� �
�
∀x ∈ (0, 2] : �
r3(x) �
�
� x �
r3(x)
x
1+ r3(x)
x
≥ − 2
3
≥ 1 − 2 1
=
3 3
≤ |x|2
6
sin x = x + r3(x) =x
≥ x · 1
> 0
3
4 2
≤ =
6 3
�
1+ r3(x)
�
x

0 ≤ y0
cos(0) = 1
cos(2) ≤ − 1
3
cos [0, 2] cos
[0, 2]
x ∈ (−π/2, 0)
sin
x − y
2
� �� �
>0
�
∀x ∈ − π π
�
, :cosx>0
2 2
π/2
∀x ∈ [0,π/2) : cos x>0
cos(x) =cos( −x
����
∈(0,π/2)
) > 0
x
−
|
−
0
−
π
2
−
π
−
3π
2
−
2π
−
cos x | 1 0 −1 0 1
sin x | 0 1 0 −1 0
exp(ix) | 1 i −1 −i 1
< 0

x = π
2
x =0
cos 0 =
sin 0 =
exp 0 =
∞�
k 02k
(−1)
(2k)!
k=0
=1
∞�
k 02k+1
(−1)
(2k +1)!
k=0
=0
∞�
k 0k
(−1)
k! =1
k=0
cos π
2 =0
2 π π
sin +cos2
2 2 =1
sin π
sin x>0 (0, 2]
= ±1
2
π/2 ∈ (0, 2]
x = π
x = 3π
2
x =2π
sin π
2 =1
exp iπ
2 =cosπ+isin
� ��
2
�
=0
π
= i
� ��
2
�
=1
cos π + i sin π = exp(iπ) =
cos 3π
2
+ i sin 3π
2
= i 2 = −1+i · 0
�
exp iπ
�2 2
= exp 3π
2
= i 3 = −i =0− i
cos 2π + i sin 2π = exp(i2π) =
= i 4 =1+0· i
�
exp iπ
�4 2

�
π
�
sin − x
2
�
π
�
cos − x
2
k ∈ Z
x ∈ R
a) cos(x +2π) =cosx
sin(x +2π) =sinx
b) cos(x + π) =− cos x
c)
sin(x + π) =− sin x
�
π
�
cosx =sin − x
�
2
π
�
sin x =cos − x
2
cos(x +2π) =cosx cos 2π
� �� �
=1
sin(x +2π) =sinx cos 2π
� �� �
=1
cos(x + π) =cosx cos π
� �� �
=−1
sin(x + π) =sinx cos π
k → k +1
= cos(−x)sin π
2
� �� �
=1
= cos(−x)cos π
2
� �� �
=0
� �� �
=−1
− sin x sin
� ��
2π
�
=cosx
=0
+cosx sin 2π
� �� �
=0
=sinx
− sin x sin
����
π = − cos x
=0
+cosx sin π
����
=0
= − sin x
− sin(−x)cos π
= cos(−x) =cosx
� ��
2
�
=0
− sin(−x)sin π
= − sin(−x) =sinx
� ��
2
�
=1
x ∈ R cos x sin x exp(ix) 2π
cos(2πk + x) = cosx
sin(2πk + x) = sinx
exp(i(2πk + x)) = exp x
cos(2π(k +1)+x) = cos(2πk +2π + x) =cos(2πk + x) =cosx
sin(2π(k +1)+x) = sin(2πk +2π + x) =sin(2πk + x) =sinx

sin(−x) =− sin x cos(−x) =cosx ∀k ∈ N
x ∈ � π
2
cos 2π
cos(2π(−k)+x) = cos(2πk − x)
= cos(−x) =cosx
sin(2π(−k)+x) = − sin(2πk − x)
= − sin(−x) =sinx
exp(i(2πk + x)) = cos(2πk + x)+i sin(2πk + x)
= cosx + i sin x =exp(ix)
�
3π , 2
a) cosx =0 ⇐⇒ x = π
+ kπ
2
k ∈ Z
b) sinx =0 ⇐⇒ x = kπ k ∈ Z
cos −π
2
cos 3π
2
= − cos π
2 =0
�
= cos π + π
�
2
= − cos π
2 =0
x = y + π y ∈ (−π/2,π/2)
cos x = cos(y + π) =− cos y < 0
����
>0
cos x>0 x ∈ � −π
2
cos x =0 x = −π
2
cos x

sin x =cos � π
2 − x�
sin x =0 ⇐⇒
⇐⇒
�
π
�
cos − x =0
2 π
π
− x = kπ +
2 2
⇐⇒ x = kπ k ∈ Z
x ∈ R
exp(ix) =1 ⇐⇒ x = k2π k ∈ Z
sin x
2
�
1
= exp
2i
ix
�
−ix
− exp
2 2
=
�=0
� �� �
exp −ix
2 (exp(ix) − 1)
2i
k ∈ Z
exp(ix) =1 ⇐⇒ sin x
2 =0
⇐⇒ x = k2π k ∈ Z
z ∈ C\{0}
a k2π k ∈ Z
z = r exp(ia) r ∈ (0, ∞),a∈ R
|z| =1 ⇐⇒ ∃a ∈ [0, 2π) : z =exp(ia)
| exp(ix)| =1 cos x sin x
{z ∈ R 2 : |z| =1}
|z| = |r exp(ia)| = |r|| exp(ia)| = r

:= |z| > 0
z
:= x + iy x, y ∈ R
|z|
� �
�
|x| ≤�
z �
�
�|z|
�
= |z|
|z| =1
cos(0) = 1
cos(π) = −1
cos b ∈ (0,π)
sin b>0
cos b = x
x 2 + y 2 = |x + iy| 2 =
� �
�
�
z �
�
�|z|
�
2
=1
y 2 = 1− x 2 =1− cos 2 b =sin 2 b
y =
a :=
� sin b y ≥ 0
− sin b y

=1
n ≥ 2,n∈ N z ∈ C
z n =1 ⇐⇒ z =exp i2kπ
n
⇒ z = r exp(ia)
1=|z n | = r n | exp(ian)| = r n
k =0,...,n− 1
z n =1 ⇐⇒ exp(ian) =1
⇐⇒ na =2kπ k ∈ Z
⇐⇒ a = 2kπ
n
k ∈ Z
exp(ix) 2π a ∈ [0, 2π)
0 ≤ k

|z| ≥1
|p(z)| C
z �= 0
C
n−1 �
p(z) = aiz i + z n
i=0
p(z) =
�
n
z
= z n
�
1+
∀i ≥ 1:
r(z) =
M :=
1+ an−1
z
n−1 �
i=0
a0
+ ...+
zn �
�
ai
z n−i
n−1 � ai
z
i=0
n−i
n−1 �
|ai|
i=0
1 1
≤
|z| i |z|
|r(z)| ≤
∀|z| ≥R := max{1, 2M}
|r(z)| ≤ 1
−
2
1
2
∀|z| ≥R
1 1
=1−
2 2
n−1 �
i=0
|ai|
|z| n−i
≤ 1
n−1 �
|ai| ≤
|z|
M
|z|
i=0
≤ r(z) ≤ 1
2
≤ 1+r(z) ≤ 1+1
2
|p(z)| = |z| n |1+r(z)|
≥ 1
2 |zn |≥ 1
2 |z|
≥ M

|p| B(0,R)
P (w)
bk �= 0
|p| p(z0) �= 0
q(w)
|p(0)| = |a0| ≤M
min |p(z)| ≤M
z∈C
P (w) = p(z0 + w)
p(z0)
P (0) = p(z0)
p(z0) =1
P (w) =1+bkw k + ...+ bnw n
z k = − 1
bk
a ∈ C a k = − 1
bk
q(w) = p(z0 + aw)
p(z0)
q(w) =1− w k + w k+1 s(w)
|s(w)| B(0, 1)
|w| ≤min � 1
2C , 1�
∀|w| ≤1: |s(w)| ≤C
|w k+1 s(w)| ≤C|w| k+1 ≤ 1
2 |w|k
|z| ≤R

p
0 0
|p(z1)|

vi = vj
v1,...,vn
vj
vj + cvi
w : V × ...× V → R
w(v1,...,vn) =0
dim V = n
w : V × ...× V → R ⇐⇒
1 ≤ j ≤ n, c ∈ R
w(v1,...,vn) =w(v1,...,vj−1,vj + cvi,vj+1,...,vn)
w(v1,...,vj−1,cvj,vj+1,...,vn) =cw(v1,...,vj−1,vj,vj+1,...,vn)
⇒
w(v1,...,vj−1,vj + cvi,vj+1,...,vn)
1.)
= w(v1,...,vn)+cw(v1,...,vi,...,vi,...,vn)
� �� �
=0
2.)
= w(v1,...,vn)

⇐ vi = vj
0 = 0· w(v1,...,vn)
ii)
= w(v1,..., 0 · vi
���� ,...,vn)
= w(v1,..., 0
���� ,...,vn)
= w(v1,..., vi − vj
� �� � ,...,vn)
i)
= w(v1,...,vj,...,vi − vj +1· vj
� �� � ,...,vn)
i)
= w(v1 ...,vn)
uj ∈ Lin(v1,...,vn) uj = � n
k=1 akvk
w(v1,...,vj−1,vj + uj,vj+1,...,vn)
i)
= w(v1,...,vj−1,vj + uj −
n�
akvk,vj+1,...,vn)
= w(v1,...,(1 + aj)vj
� �� � ,...,vn)
k=1,k�=j
ii)
= (1+aj) · w(v1,...,vj,...,vn)
ii)
= w(v1,...,vj,...,vn)+w(v1,...,ajvj,...,vn)
i)
= w(v1,...,vj,...,vn)+w(v1,...,vj−1,ajvj +
= w(v1,...,vj,...,vn)+w(v1,...,uj,...,vn)
n�
k=1,k�=j
vj ∈ Lin(v1,...,uj,...,vn)
uj �∈ Lin(v1,...,vn) vj �∈ Lin(v1,...,uj,...,vn)
dim Lin(v1,...,vj−1,vj+1,...,vn) =n − 1
dim Lin(v1,...,vn) = n
dim Lin(v1,...,vn,uj) = n +1
akvk,vj+1,...,vn)

dim V = n
dim Lin(v1,...,vj−1,vj+1,...,vn) ≤ n − 2
dim Lin(v1,...,vn) ≤ n − 1
dim Lin(v1,...,vj−1,uj,vj+1,...,vn) ≤ n − 1
dim Lin(v1,...,vj−1,uj + vj,vj+1,...,vn) ≤ n − 1
w(v1,...,vj−1,uj + vj,vj+1,...,vn)
= 0 = 0+0
= w(v1,...,vn)+w(v1,...,vj−1,uj,vj+1,...,vn)
v1,...,vn
vj = � n
i=1,i�=j civi
w(v1,...,vn) =0
w(v1,...,vj,...,vn)
n�
= w(v1,...,vj − civi,...,vn)
i=1,i�=j
= w(v1,...,0,...,vn)
= 0
vi
vj
w(v1,...,vi,...,vj,...,vn) =−w(v1,...,vj,...,vi,...,vn)
w(v1,...,vn)
i)
= w(v1,...,vi,...,vj − vi,...,vn)
i)
= w(v1,...,vi + vj − vi,...,vj
− vi,...,vn)
� �� �
=vj
1.)
= w(v1,...,vj,...,−vi,...,vn) − w(v1,...,vj,...,vj,...,vn)
� �� �
=0
1.)
= −w(v1,...,vj,...,vi,...,vn)

−1
{1,...,n}
Sn := {r : {1,...,n}→{1,...,n},r }
r ∈ Sn
R : W → W, (i, j) ↦→
r −1 (i) r −1 (j)
r(i) �= r(j)
W := {(i, j) :1≤ i

�
1≤i

t ′ ∈ Sn
t ∈ Sn
r ∈ Sn
s(t ′ ) = �
s(t) =s(t −1 )
s(t) =−1
t = r −1 t ′ r
i ↦→ 1,j ↦→ 2
3,...,n t ′ (j) =j 3,...,n
t
1≤i

a21 �= 0
a12 �= 0
a22 �= 0
a11 �= 0
⇐⇒ ∃c ∈ K : c(a11,a12) =(a21,a22)
⇐⇒ ∃c ∈ K : c = a21
a11
⇐⇒ a21
a12 = a22
a11
⇐⇒ a11a22 − a12a21 =0
ca12 = a22
⇐⇒ ∃c ∈ K : (a11,a12) =c(a21,a22)
⇐⇒ ∃c ∈ K : c = a11
a21
⇐⇒ a11
a22 = a12
a21
⇐⇒ a11a22 − a12a21 =0
a12 = ca22
⇐⇒ ∃c ∈ K : c(a11,a12) =(a21,a22)
⇐⇒ ∃c ∈ K : c = a22
a12
⇐⇒ a22
a11 = a21
a12
⇐⇒ a11a22 − a12a21 =0
ca11 = a21
⇐⇒ ∃c ∈ K : (a11,a12) =c(a21,a22)
⇐⇒ ∃c ∈ K : c = a12
a22
⇐⇒ a12
a21 = a11
a22
⇐⇒ a11a22 − a12a21 =0
a11 = ca21
Mn(K) n × n K

⎛
⎜
det : Mn(K) → K, ⎝
�
t∈Sn
= �
p ∈ Sn
t∈Sn
det : Mn(K) → K
a11 ··· a1n
an1 ··· ann
det(1n) =1
⎞
⎟
⎠ ↦→ �
t∈Sn
s(t)a 1t(1) · ...(a it(i) + cb it(i)) ...· a nt(n)
s(t)a 1t(1) · ...· a nt(n)
s(t)a1t(1) · ...ait(i) ...· ant(n) + c �
s(t)a1t(1) · ...bit(i) ...· ant(n) Sn
t∈Sn
∀1 ≤ i ≤ n : aik = aij
p 2 = id
{t, t ′ }
pt ′ = t ⇐⇒ p 2 t ′ = pt p2 =id
⇐⇒ t ′ = pt
t(i) =j, k
a i,t(i) = a i,pt(i)
a i,p(j) = ai,k = aij
a i,p(k) = ai,j = aik
a i,t(i) = a i,pt(i)
j �= t(i) �= k
1 ≤ i ≤ n
j �= k

s(t)a 1,t(1) ···a n,t(n) + s(p ◦ t)a 1,pt(1) ···a n,pt(n)
= s(t)a 1,t(1) ···a n,t(n) + s(p)s(t)a 1,t(1) ···a n,t(n)
= s(t)(1 + s(p))a1t(1) ...ant(n) = s(t)(1− 1) a1t(1) ...ant(n) = 0
1n
t ∈ Sn
� �� �
=0
i �= j aij =0
t = id
V := {(t(i),i):1≤ i ≤ n}
a) (t −1 ,t −1 ):V → V,(t(i),i) ↦→ (i, t −1 (i))
b) detA T =detA
t −1 (i) ∈{1,...,n}
(i, t −1 (i)) = (t(t −1 (i)),t −1 (i)) ∈ V
(i, t −1 (i)) = (j, t −1 (j)) ⇐⇒ i = j t −1 (i) =t −1 (j)
(t −1 ,t −1 )=V (t(i),i)
t −1
⇐⇒ i = j
(t −1 ,t −1 )(t(t(i)),t(i)) = (t(i),i)
a t(1),1 ···a t(n),n = �
(i,j)∈V
aij = a 1,t −1 (1) ···a n,t −1 (n)
t t −1 Sn t −1

det A T
det A =detA T
Ae1,...,Aen
Ae1,...,Aen
=
s(t)=s(t −1 )
=
=
�
s(t)at(1),1 ···at(n),n t∈Sn
�
t∈Sn
�
t∈Sn
= detA
s(t −1 )a 1,t −1 (1) ···a n,t −1 (n)
s(t)a 1,t(1) ···a n,t(n)
det Eij,cA =detA =detAEij,c
det AEij,c = det(a1,...,aj + cai,...,an)
= det(a1,...,an) =detA
det Eij,cA = det(Eij,cA) T =detA T Eji,c
= detA T =detA
w : Mn(K) → K
Mn(K) w(1n) =c
w = c · det
w(A) =0=c · det(A)
� �� �
=0
E T m ...E T 1 A T =
⎛
⎜
⎝
Eij,c
a ′ 11
0
0 a ′ nn
A T
⎞
⎟
⎠
Eij,c

(n − 1)
AE1 ...Em = (AE1 ...Em) TT
det w
= (E T m ...E T 1 A T ) T
=
⎛
⎜
⎝
a ′ 11
0
0 a ′ nn
⎛
⎜
det A = det(AE1 ...Em) =det⎝
=
� n�
i=1
a ′ ii
�
· det 1n =
n�
i=1
⎛
⎜
w(A) = w(AE1 ...Em) =w ⎝
= w(1n)
= c det A
det nA =
w(A) =
n�
i=1
a ′ ii = c
1 ≤ i ≤ n
n�
i=1
a ′ ii
a ′ ii
a ′ 11
a ′ 11
⎞
⎟
⎠
0
0 a ′ nn
0
0 a ′ nn
n�
(−1) i+j aij det n−1Aij
j=1
n�
(−1) i+j aij det n−1Aij
j=1
Aij
⎞
⎟
⎠
⎞
⎟
⎠
(n − 1) ×

det n−1
=
k �= j
n�
(−1) i+j (aij + bcij)detn−1Aij
j=1
n�
(−1) i+j n�
aij det n−1Aij + b (−1) i+j cij det n−1Aij
j=1
j=1
k l k

0
det
w(1n) = (−1) i+i · 1 · det n−1Aii
det nA =
= 1
w =det
n�
(−1) i+j aij det n−1Aij
i=1
det nA = detnA T =
=
=
n�
(−1) i+j a T ij det n−1(A T )ij
j=1
n�
(−1) i+j aji det n−1Aji
j=1
n�
(−1) i+j aij det n−1Aij
i=1
r =0 det B =detB
� 1r C
0 B
�
�
1r C
det
0 B
�
1r−1 C
=det
′
0 B
�
A C
det
0 B
�
�
=detB
�
′′ 1 C
= ...=det
0 B
�
=detA · det B
�
=detB

�
A C
w : Mr(K) → K,A↦→ det
0 B
� A C
0 B
�
�
A C
w(A) =det
0 B
�
1r C
w(1r) =det
0 B
�
A C
det
0 B
�
=0
�
=detB
�
=detA · det B
⎛
A1
⎜ 0
det ⎜
⎝
∗
A2
···
···
···
∗
∗
⎞
⎟
⎠
0 0 ··· Am
=detA1 ···det Am
A, B ∈ Mn(K)
(m − 1)
det(AB) =detA · det B
w : Mn(K) → K,B ↦→ det AB =det(Ab1,...,Abn)
bi = bj Abi = Abj AB
w(B) =detAB =0
�
det AB

det A �= 0
(Ae1,...,Aen)
w(1n) =detA1n =detA
w(AB) =detA det B
det(A −1 )=(detA) −1
1=det1n =detAA −1 =(detA)(det A −1 )
⎛
⎜
det ⎝
a11
0
∗ ∗
∗
⎞
⎟
⎠ =
0 0 ann
n�
i=1
aii
⎛
a11
⎜
det ⎝ 0
∗ ∗
∗
⎞
⎟
⎠
=
0 0
⎛
a22
⎜
a11 det ⎝ 0
ann
∗ ∗
∗
⎞
⎟
⎠
= ...
0 0 ann
= a11 ...ann

det A �= 0 ⇐⇒
u1,...,un
⎛
⎜
Em ...E1A = ⎝
a ′ 11
Eij,c
∗
0 a ′ nn
det A = detEm ...E1A
⎛
a
⎜
= det⎝
′ 11
=
n�
i=1
n�
a ′ ii
∗
0 a ′ nn
a ′ ii �= 0
i=1
⇐⇒ ∀1 ≤ i ≤ n : a ′ ii �= 0
⇐⇒ a ′ 1,...a ′ n
⎞
⎟
⎠
⎞
⎟
⎠
En...E1 ⇐⇒ a1,...,an
⇐⇒ A
⇐⇒ A
f : V → V
v1,...,vn
det A =detS −1 A ′ S =detS −1 det A ′ det S =detA ′
det f := det A
Ãij
ãij =(−1) i+j det Aji
AÃ =detA · 1n

det A �= 0
Aj(b)
det nA =
A −1 = Ã
det A
n�
(−1) i+j aij det n−1Aij =
j=1
⎛
⎜
= (ai1,...,ain) ⎝
ã1i
ãni
⎞
⎟
⎠
n�
j=1
aijãji
= (i A) · (i Ã)
= (AÃ)ii 0 = detB =
=
=
det B =0
n�
(−1) k+j bkj det n−1Bkj
j=1
n�
(−1) k+j aij det n−1Akj
j=1
n�
j=1
aijãjk
⎛
⎜
= (ai1,...,ain) ⎝
ã1k
ãnk
⎞
⎟
⎠
= (i A) · (k Ã)
= (AÃ)ik xj =
det Aj(b)
det A
Ax = b

Aj(b)
det Aj(b) =
x = A −1 b = Ãb
det A
=
n�
aij (−1)
����
i+j det n−1Aij
� �� �
i=1
n�
i=1
=bi
ãjibi
⎛
⎜
= (ãj1,...,ãjn) ⎝
= ( Ãb)j
= xj · det A
ãji
b1
bn
⎞
⎟
⎠

V = W
V �= W f : V → W
� 1r 0
0 0
dim V = n vr+1,...,vn
v1,...,vn
f(v1),...,f(vr)
dim Bild(f)+dimNull(f) =dimV
f(v1),...,f(vr)
�
f(v1),...,f(vr),wr+1,...,wm
f(vi) = f(vi) 1 ≤ i ≤ r
f(vi) = 0 r +1≤ i ≤ n
� 1r 0
0 0
�
f : V → V c ∈ K 0 �= v ∈ V
f(v) =cv
{v ∈ V |f(v) =cv}
f(0) = 0 = c · 0
f(v1 + v2) = f(v1)+f(v2) =cv1 + cv2 = c(v1 + v2)
f(bv) = bf(v) =bcv = c(bv)

− 1 → r
vi
v1 ...,vr−1
cr �= ci
c1,...,cr
r =1 bv1 =0 v1 �= 0
f
� r
i=1 aivi =0
cr
0=f(0) = f
0=
� r
i=1 aivi
r�
i=1
� r�
i=1
craivi −
0=
aivi
r�
i=1
b =0
r�
i=1
�
=
craivi
r�
aif(vi) =
i=1
r−1
aicivi =
r�
i=1
aicivi
�
(cr − ci)aivi
i=1
∀1 ≤ i ≤ r − 1: ai (cr − ci) =0
� �� �
�=0
arvr =0
∀1 ≤ i ≤ r − 1: ai =0
ar =0
dim V = n f : V → V
c ∈ R
pf (x) =det(x · idV − f)
f : V → V
v1,...,vr

x =0
det f =detA
f : V → V ⇐⇒
pf (c) =0
⇐⇒
pf (c) =0
det(c · idV − f) =0
⇐⇒ c · idV − f
⇐⇒ ∃v �= 0:(c · idV − f)(v) =0
⇐⇒ ∃v �= 0:f(v) =c · v
pA(x) =x n − x n−1 ·
id ∈ Sn
n�
aii + ...+(−1) n det A
i=1
det(x · id − f)
(x − a11) · ...· (x − ann) =x n − x n−1
f : V → V
n − 2
x n x n−1
n�
aii + ...
i=1
pA(0) = det(−A)
= det(−a1,...,−an)
= (−1) n det A
dim U = r
pf ≥ r

pf
c
u1,...,ur
u1,...,ur,ur+1,...,un
f(ui) =cui
1 ≤ i ≤ r
⎛
⎜
⎝
c
0
0
c
∗
∗
∗
⎞
⎟
⎠
0 B
⎛
x − c 0 0 ∗
⎞
pf (x) =
⎜
det ⎜
⎝
0 0
x − c
∗
∗
⎟
⎠
0 0 x1n−r − B
= (x−c) r det(x1n−r − B)
dim U
A =
� c 1
0 c
�
Av = cv ⇐⇒
⇐⇒
⇐⇒
(cidV − A)v =0
� �� �
0 1 v1
=0
0 0 v2
� �
v2
=0
0
⇐⇒ v2 =0
�
� 1
0
�
1
U := Lin{v ∈ V : f(v) =cv} = K
0
dim U = 1
pf
�

pf = det(x · idV − A)
� �
x − c −1
= det
0 x − c
= (x − c) 2
2= > dim U =1
U1 ...,Ur
f : V → V c1,...,cr
uij
u11,...,u1k1 ,...,ur1,...,urkr
kj �
j=1
r� kr �
i=1 j=1
aijuij =0
aijuij =0 1 ≤ i ≤ r
aij =0 1 ≤ j ≤ kj
aij =0 1 ≤ i ≤ r, 1 ≤ j ≤ kj
f : V → V dim V = n
⇐⇒ pf (x)
⇒ v1,...,vn
⎛
⎜
A = ⎝
a11
0
0 0
0
⎞
⎟
⎠
0 0 ann
ui1,...,uiki

vi
vi
aii
aii
f(vi) =aiivi
⎛
⎜
det(x · idV − A) =det⎝
x − a11
0
0 0
0
⎞
⎟
⎠ =
0 0 x − ann
c1 k1
c1 k1 v1,...,vk1
a11 = ...= ak1k1 = c1 �= ak1+1,k1+1
c2,...,cr
pf (x)
n�
(x − aii)
ci ki ki
⇐ pf C
r�
n =
=
⎛
⎜
⎝
i=1
r�
i=1
Ui
ci
u11,...,u1k1 ,...,ur1,...,urkr
c1
f(uij) =ciuij
c1
cr
0 cr
0
Ui
⎞
⎟
⎠
i=1

C
R
�1 i
�
�=0
�
,
�� 1
i
c ∈ C,u,v,w ∈ V
C n
C
〈v, v〉 =0⇒ v =0
� ��
1
i R2 C
=1 2 + i 2 =1− 1=0
〈·, ·〉 : V × V → C
1.) 〈cv, w〉 = c 〈v, w〉
2.) 〈v, cw〉 = c 〈v, w〉
3.) 〈u + v, w〉 = 〈u, w〉 + 〈v, w〉
4.) 〈u, v + w〉 = 〈u, v〉 + 〈u, w〉
3.) 〈v, w〉 = 〈w, v〉
4.) 〈v, v〉 > 0 v �= 0
⎛
⎜
〈a, b〉 := (a1 ...,an) ⎝
〈a + a ′ ,b〉 =
=
b1
bn
C
⎞
n�
⎟
⎠ =
n�
(ai + a ′ i)bi
i=1
n�
i=1
aibi +
n�
i=1
= 〈a, b〉 + 〈a ′ ,b〉
i=1
a ′ ibi
aibi

〈a, b + b ′ 〉 =
〈ca, b〉 =
〈a, cb〉 =
〈a, b〉 =
〈a, a〉 =
v1,...,vn
〈a, a〉 =
n�
i=1
i=1
=
n�
i=1
n�
i=1
ai(bi + b ′ i )
aibi +
n�
i=1
= 〈a, b〉 + 〈a, b ′ 〉
aib ′
i
n�
n�
(cai)bi = c aibi = c 〈a, b〉
i=1
i=1
n�
n�
ai(cbi) =c aibi = c 〈a, b〉
n�
i=1
aibi =
n�
i=1
〈vi,vj〉 =
aiai =
i=1
n�
aibi = 〈b, a〉
i=1
n�
|ai| 2 ≥ 0
i=1
|ai| 2 =0 ⇐⇒ ∀i : ai =0
⇐⇒ a =0
v, w ∈ V ⇐⇒
〈v, w〉 =0
⇐⇒
〈v, v〉 =1
� 1 i = j
0 i �= j
⇐⇒

w1,...,wn
C 〈·, ·〉 dim V = n
v1,...,vn
j=1
v =
n�
〈v, vi〉 vi
i=1
v = �n j=1 ajvj
�
n�
�
〈v, vi〉 = ajvj,vi =
v1,...,vn
n�
aj 〈vj,vi〉 = ai
j=1
∀1 ≤ i ≤ n : Lin(v1,...,vi) =Lin(w1,...,wi)
w1 :=
v1
�
〈v1,v1〉
�i−1
∀i ≥ 2: ui := vi −
∀i ≥ 2: wi :=
j=1
ui
� 〈ui,ui〉
i =1: v1 �= 0
〈w1,w1〉 =
=
〈vi,wj〉 wj
�
v1
�
〈v1,v1〉 ,
�
v1
�
〈v1,v1〉
〈v1,v1〉
� 2 =1
〈v1,v1〉
i − 1 → i v1,...,vn ui �= 0
〈ui,ui〉 > 0
wi =
ui
� 〈ui,ui〉
v ∈ V
� 〈v1,v1〉 > 0 v1

k

�i−1
ui := vi − 〈vi,wj〉 wj ∈ Lin(v1,...,vi)
wi :=
vi = wi
j=1
ui
� ∈ Lin(v1,...,vi)
〈ui,ui〉
� �i−1
〈ui,ui〉 + 〈vi,wj〉 wj ∈ Lin(w1,...,wi)
j=1
C dim V = n
v1,...,vn v = �n i=1 aivi,w = �n i=1 bivi
〈v, w〉 V
〈v, w〉 V = 〈a, b〉 C n
f : C n → V
〈f(v),f(w)〉 V = 〈a, b〉 C n
=
=
=
�
n� n�
aivi,
i=1
n�
i=1 j=1
n�
i=1
n�
aibi
= 〈a, b〉 C n
j=1
bjvj
�
aibj 〈vi,vj〉 V
f : {e1,...,en} →{v1,...vn},ei ↦→ vi
f : C n → V
V
v1,...,vn

f(e1),...,f(en)
〈f(v),f(w)〉 V
=
=
Def
=
=
� �
n�
f
n�
i=1
i=1 j=1
n�
n�
i=1 j=1
n�
i=1
n�
aiei
� ⎛
n�
,f⎝
cjej
j=1
aicj 〈f(ei),f(ej)〉 V
aicj 〈vi,vj〉 V
aici = 〈v, w〉 C n
⎞�
⎠
V

R C R
A A
⇐
A : K n → K n ⇐⇒
∀v, w ∈ K n : 〈Av, Aw〉 = 〈v, w〉
⇐⇒
A T A =1n
A −1 = A T
A T A = 1 = AA T
〈Av, Aw〉 = (Av) T Aw = v T A T Aw
= v T 1nw = v T w = 〈v, w〉
⇒ v = ei w = ej B = A T A
bij = e T i Bej = e T i A T Aej
= (Aei) T =
Aej = 〈Aei,Aej〉
�
1 i = j
〈ei,ej〉 =
0 i �= j
A T A =1n
0=〈Av, Av〉 = 〈v, v〉 �= 0
v �= 0 Av =0
A −1 A T A =1
A −1 = A T
A T A = 1 = AA T

⇐
(a1,...,an)
⇒
( ) · ( )
= 〈ai,aj〉 =(Aei) T (Aej)
= e T i A T Aej = e T =
i ej
�
1 i = j
0 i �= j
( ) · ( )
= � A T ei,A T � T
ej =(A ei) T (AT ej)
= e T i AA T ej = e T i ej
�
1 i = j
=
0 i �= j
A = A T ⇐⇒ ∀v, w ∈ K n : 〈Av, w〉 = 〈v, Aw〉
〈Av, w〉 = (Av) T w = v T A T w
A=A T
= v T Aw = 〈v, Aw〉
aij = e T i A ej
����
=ej
= 〈ei,Aej〉
= 〈Aei,ej〉 =(Aei) T ej
= aij
A : C n → C n
A = A T
A : R n → R n
A = A T

c ∈ C
c 〈v, v〉 = 〈cv, v〉 = 〈Av, v〉 A=AT
= 〈v, Av〉
= 〈v, cv〉 = c 〈v, v〉
c = c
c ∈ R
n =1 f : C → C f(x) =cx c = f(1)
c
n → n +1 C
c ∈ C c ∈ R
u1
U1 ∩ U2 = {0}
v ∈ U1 ∩ U2
v1 :=
U1 := Cv1
u1
� 〈u1,u1〉
U2 := {w ∈ V |〈w, v1〉 =0}
0
Null(f) =U2
v = av1
v∈U2
= 〈v, v1〉 = a 〈v1,v1〉
� �� �
�=0
0 = a
0 = av1 = v
f : V → C,w ↦→ 〈w, v1〉
Bild(f) =C
f(v1) = 〈v1,v1〉 =1�= 0
f(Cv1) = C
dim U2 = dimNull(f)
= dimV−dim Bild(f)
= n − dim C = n − 1

U1 + U2 = V
U2
Av1 = cv1 ∈ U1
〈Au2,v1〉 A=AT
= 〈u2,Av1〉
= 〈u2,cv1〉 = c 〈u2,v1〉 =0
AUi ⊂ Ui
∀2 ≤ j ≤ n : v1⊥vj v1,...,vn
A = A T
A : R n → R n
A : R n → R n
A : C n → C n
A T = A T = A
A = A T
AA T = A T A C
v2,...,vn
v ∈ V
v ∈ V c A T
(A − cidV )(A T − cidV )
= AA T − c(A + A T )+ccidV
= (A T − cidV )(A − cidV )
n−1

0 = 〈(A − cidV )v, (A − cidV )v〉
�
= (A T �
− cidV )(A − cidV )v, v
�
= (A − cidV )(A T �
− cidV )v, v
�
= (A T − cidV )v, (A T �
− cidV )v
(A − cidV )(v) =0 ⇐⇒ (A T − cidV )(v) =0

⇐⇒ l, m, n ∈ R
m + n = n + m
l +(m + n) =(l + m)+n
∃0 ∈ R :0+m = m +0=m
∀m ∈ R∃−m ∈ R : m +(−m) =0
(mn)l = m(nl)
l(m + n) =lm + ln
⇐⇒
mn = nm
∀b ∈ I ∀r ∈ R : rb, br ∈ I
∀b, b ′ ∈ I : b + b ′ ∈ I
0 ∈ I
∀x ∈ K
∃1 ∈ R ∀m ∈ R :1· m = m · 1=m
⇐⇒
∀m ∈ R ∃m −1 ∈ R : mm −1 =1
�
n�
K[x] := aix i �
: ai ∈ K,n∈ N
i=0
p(x) ≡ 0 ∈ K[x] P
−p(x) ∈ K[x] P
p(x) ≡ 1 ∈ K[x] P
p1(x)+p2(x)
K
K
= p2(x)+p1(x)
I ⊂ R ⇐⇒

ak �= 0
p = �n i=0 aixi q = �m i
i=0 bix
s, r
an �= 0�= bm
�n i
i=0 aix
p = sq + r Grad r < Grad q r =0
Grad p < Grad q : s =0 r = p
Grad p ≥ Grad q :
Grad p = Grad q = n :
p = sq + r =0+p = p
Grad r
p=r
= Grad p < Grad q
s := an
bn
r := p − sq =
=
n−1 �
�
i=0
(Grad p) → (Grad p)+1=n :
Grad q = m

Grad t < (Grad p)+1 s ′ ,r ′
t = qs ′ + r ′
Grad r ′ < Grad q r ′ =0
p =
= anx n−m
=
anx n−m
q + t
bm
q + s
bm
′ q + r ′
�
anxn−m + s ′
�
q + r ′
bm
0 �= I ⊂ K[x]
m = x d �d−1
+
p ∈ I q ∈ K[x]
0 �= I
Grad r < d
i=0
p = qm
aix i
d := min{s : s = Grad p, p ∈ I}
m ∈ I Grad m = d ad
p ∈ I
r =0
p = qm + r r =0 Grad r < d
r = p − qm ∈ I
p = qm

Grad r < d
m m ′
m, m ′
m = am ′ + r r =0 Grad r < d
r = m − am ′ ∈ I
m = am ′
m m ′
Grad m = Grad m ′ + Grad a
d = d + Grad a
a ∈ K
a = 1
m = m ′

K dim V = n f : V → V
pf
v �= 0
Wi = Lin(v, f(v),f 2 (v),...,f i (v))
a) Wi ⊂ Wi+1
b) ∃r ≤ n : Wr−1 = Wr
c) ∀k ≥ r : Wk = Wr−1
v,...,f r (v)
f : V → V
Wi = Lin(v, f(v),f 2 (v),...,f i (v))
⊂ Lin(v, f(v),f 2 (v),...,f i+1 (v))
= Wi+1
r ∈ N
f r (v) ∈ Lin(v,...,f r−1 (v))
det(c · idV − f)
dim V = n r ≤ n v,...,f n (v)
dim V = n
k = r :
k → k +1: Wk = Wr−1
Lin(v,...,f r−1 (v)) = Lin(v,...,f r (v))
v, f 1 (v),...,f k (v) ∈ Wr−1
f(v),...,f(f k (v)) ∈ Wr = Wr−1
Wk+1 = Lin(v,...,f k+1 (v)) ⊂ Wr−1
Wk+1 ⊂ Wr−1 ⊂ Wk+1

� n
i=0 aif i : V → V
1.)
2.)
{f : V → V |f }
1.) f(v)+g(v) =g(v)+f(v)
2.) f(v)+(g(v)+h(v)) = (f(v)+g(v)) + h(v)
3.) 0 : V → V,v ↦→ 0
4.)
f(v)+0=0+f(v)
f : V → V − f : V → V,v ↦→ −f(v).
5.)
f(v)+(−f(v)) = 0 = −f(v)+f(v)
(fg)h(v) =f(g(h(v))) = f(gh(v))
6.) f(g(v)+h(v)) = f(g(v)) + f(h(v))
7.) id : V → V,v ↦→ v
i=0
f(id(v)) = f(v) =id(f(v))
m�
aif i n�
(v)+ bjf j n�
(v) = bjf j m�
(v)+ aif i (v)
i=0
j=0
⎛
m�
⎝ aif i ⎛
n�
+ ⎝ bjf j +
j=0
⎛⎛
m�
= ⎝⎝
aif i +
i=0
n�
j=0
bjf j
j=0
l�
i=0
⎞
⎠ +
cif i
l�
i=0
3.) 0 : V → V,v ↦→ 0
m�
aif i m�
(v)+0=0+ aif i (v)
i=0
i=0
⎞⎞
⎠⎠
(v)
cif i
i=0
⎞
⎠ (v)

4.)
5.)
6.)
i=0
m�
aif i (v) :V → V
i=0
i=0
m�
(−ai)f i (v) :V → V,v ↦→ −f(v).
m�
aif i m�
(v)+ (−ai)f i m�
(v) =0= (−ai)f i m�
(v)+ aif i (v)
⎛
m�
⎝ aif i ◦
i=0
i=0
n�
j=0
bjf j
⎞
⎠ ◦
m�
= aif i ⎛
n�
◦ ⎝ bjf j ◦
m�
i=0
i=0
aif i
j=0
⎛
n�
⎝ bjf j +
j=0
j=0
l�
k=0
i=0
l�
ckf k (v)
k=0
l�
k=0
ckf k
ckf k
i=0
⎞
⎠ (v)
⎞
⎠ (v) =
m�
aif i n�
◦ bjf j m�
(v)+ aif i ◦
i=0
l�
ckf k (v)
k=0
7.) id : V → V,v ↦→ v
m�
aif i m�
(id(v)) = aif i �
m�
(v) =id aif i �
(v)
8.)
i=0
i=0
j=0
i=0
i=0
j=0
i=0
j=0
i=0
m�
aif i n�
◦ bjf j m� n�
= ◦ aif i bjf j n�
= bjf j m�
◦
f : V → V
J : K[x] →{f : V → V },
n�
aix i ↦→
i=0
n�
i=0
i=0
aif i
aif i
{0} �= Null(J) ⊂ K[x] mf

i=0
a, b ∈ K
�
m�
J a aix i m�
+ b
i=0
J(0) = 0 ◦ f =0
� �
m�
= J (aai + bbi)x i
�
bix i
i=0
i=0
m�
= a aif i m�
+ b
= aJ
� m�
i=0
aix i
i=0
bif i
� �
m�
+ bJ
i=0
dim{f : V → V |f } = dim{n × n − }
= n 2
∞ =dimK[x] =dimNull(J)+dimBild(J)
� �� �
≤n 2
∞ = dimNull(J)
{0} �= Null(J)
�m i=0 aixi , �m i=0 bixi ∈ Null(J)
�m i=0 cixi ∈ K[x]
�
m�
J aix i m�
·
i=0
i=0
i=0
�
m�
J cix i m�
·
i=0
�
m�
J aix i m�
+
i=0
i=0
cix i
aix i
bix i
�
�
�
=
=
=
m�
aif
i=0
i
� ��
≡0
�
m�
cif i ◦
i=0
m�
aif
i=0
i
� ��
≡0
�
J(0) ≡ 0
i=0
bix i
�
m�
◦ cif i m�
=0◦ cif i ≡ 0
m�
i=0
aif i
� ��
≡0
�
m�
+ bif i
i=0
� �� �
≡0
=
≡ 0
i=0
m�
cif i ◦ 0 ≡ 0
i=0

ci ∈ K
i =1,...,n
f : V → V
V0,...,Vn
{0} = V0 ⊂ V1 ⊂ ...⊂ Vn = V
dim Vi = i
f(Vi) ⊂ Vi
C
pf =(t − c1) ···(t − cn)
⇒ v1 ∈ V1 V1 = Kv1
Vi vi+1 Vi+1
A =(f(v1),...,f(vn))
f(Lin(v1,...,vi)) = f(Vi) Vor
⊂ Vi = Lin(v1,...,vi)
⇒ Vi = Lin(v1,...,vi)
⇒
f(Vi) ⊂ Vi
pA(t) =
=
det(t · idV − A)
⎛
t − a11
⎜
det⎝
∗
⎞
⎟
⎠
0 t − ann
n�
= (t − aii)
i=1
⇒ n =1 A : K → K
C

n − 1 → n
pf (c1) =0 v1 c1
(v1,w2,...,wn)
f(V1) ⊂ V1
V = Kv1 ⊕ Lin(w2,...,wn) =V1 ⊕ W
A =
=
cE11 + H + G
⎛
c1 0 ...
⎜ 0 0 ...
⎜
⎝
⎞
0
0 ⎟
⎠
0 0 ... 0
+
⎛
⎜
⎝
0
0
a12
0
...
...
a1n
0
⎞
⎟
⎠
⎛
0 0 ...
⎜ 0 a22 ...
+ ⎜
⎝
0
a2n
0
⎞
⎟
⎠
0 ... 0
0 an2 ... ann
(Wi)i
pA(t) = det(t · idV − A)
= (t−c1)det(t · idV − G)
= (t−c1) · pG(t)
pG =(t − c2) ···(t − cn)
G : W → W dim W = n − 1
{0} = W0 ⊂ ...⊂ Wn−1 = W
G(Wi) ⊂ Wi
dim Wi = i
{0} = V0 ⊂ V1 ⊂ V1 ⊕ W1 ⊂···⊂V1 ⊕ Wn−1 = V
bv1 + wi ∈ V1 ⊕ Wi
f(bv1 + wi) = c1bv1 + H(wi) + G(wi) ∈ V1 ⊕ Wi
� �� � � �� �
∈V1
f(V1 ⊕ Wi) ⊂ V1 ⊕ Wi
∈Wi

v1 �∈ Wi
f(V1) ⊂ V1
f|W
dim V1 ⊕ Wi = i +1
f : V → V 0 �= v ∈ V W ⊂ V
W = Lin(v, f(v),...,f r−1 (v))
⎛
⎜ 0 1
A|W = ⎜ 0
⎜
⎝
0 0 ... ... 0 −a0
1 0 ... ... 0 −a1
1 0 −ar−2
0 0 ... 0 1 −ar−1
A|W
J(p A|W )(v) =f r (v)+ar−1f r−1 (v)+...+ a1f(v)+a0v =0
⎞
⎟
⎠
f i (v) W
f i+1 (v)
f r (v) v, f(v),...,f r−1 (v) ai
f r �r−1
(v) = (−ai)f i (v)
i=0
=
det(t · idW − f|W )
⎛
t 0 ...
⎜
−1 t ...
⎜ 0 −1
det ⎜ 0
⎜
⎝
...
...
−1
0
0
t
a0
a1
ar−2
⎞
⎟
⎠
0 0 ... 0 −1 t + ar−1

(−1) r (−1) 1 ⎛
−1 t ... ... 0
⎞
⎜ 0
⎜
a0 det ⎜
⎝
−1
0
−1 t
⎟
⎠
0 0 ... 0 −1
+(−1) r (−1) 2 ⎛
t 0 ... ... 0
⎞
⎜ 0
⎜
a1 det ⎜
⎝
−1
0
t
−1 t
⎟
⎠
0 0 ... 0 −1
+ ...+(−1) r (−1) r ⎛
t 0 ... ...
⎞
0
⎜ −1
⎜
(t + ar−1)det ⎜ 0
⎜
⎝
t
−1
0
t
⎟
0 ⎠
0 0 ... −1 t
(−1) r (−1) i
p(A|W ) = a0 + a1t + ...+(t + ar−1)t r−1
= t r + ar−1t r−1 + ...+ a0
J(p A|W )(v) = f r (v)+ar−1f r−1 (v)+...+ a0v =0
f : V → V W ⊂ V
f(W ) ⊂ W
pf|W |pf
pf ∈ Null(J)
mf |pf
r
0 �= v1 ∈ W v f
U1 �= W 0 �= v2 ∈ W \U1
U1 = Lin(v, f(v) ...,f r−1 (v))
W = U1 ⊕ ...⊕ Uk

v ∈ V
V = U1 ⊕ ...⊕ Uk ⊕ Z
⎛
⎞
A|U1 0 ∗
⎜
⎟
⎜
⎟
A = ⎜
⎟
⎝ 0 A|Uk
⎠
0 B
pf (t) = det(t · idV − f)
= det(t · idZ − B)
k�
det(t · idUi − A|Ui )
i=1
= det(t · idZ − B) · p f|W
p f|W
| pf
v ∈ V W f v
W = Lin(v, f(v),...,f r (v))
J(pf|W )(v) =0∈ V
J(pf )(v) =J(pf|Z ) ◦ J(pf|W )(v) =0
� �� �
=0∈V
pf ∈ Null(J) mf
∀v ∈ V : J(pf )(v) = 0
J(pf ) ≡ 0
pf ∈ Null(J)
mf |pf

f : V → V dim V = n
Grad mf ≤ n
pf
pf
pf (t)
ci
m n f
mf |pf
C
mf (ci) vi
� �� �
∈C
pf = mf · qf
Grad pf = Gradmf · Gradqf
Grad mf ≤ Grad pf = n
ci
pf =(t − c1) r1 ···(t − ck) rk
vi ci f(vi) =civi
=
=
mf =
= 0
k�
j=0
⎛
k�
⎝
bjx j
⎞
bjc
j=0
j⎠
vi
⎛
k�
⎝
vi �= 0 ci mf
⎞
bjf
j=0
j⎠
vi = J(mf ) (vi)
� �� �
≡0:V →V
(ci − t) | mf (t)
mf (t) pf (t)
pf |m n f
f : V → V dim V = n
1 ≤ d ≤ n f d ≡ 0

pf (t) =t n
pf
⇒
f =
⇒ f d ≡ 0
mf
pf = t n
⎛
⎜
⎝
pf = t s
0 ∗ ∗
∗
0 0
t d ∈ Null(J)
mf |t d
pf = t n
⎞
⎟
⎠
d ≤ s ≤ n
pf =det(R − tE) =(t − r11) ···(t − rnn)
⇒ f 2 ,f 3 ,...
∀1 ≤ i ≤ n : rii =0
f 2 ⎛
⎞
⎛
⎞2
0 0 ∗ ··· ∗
0 ∗ ... ∗ ⎜
⎟
⎜
⎟ ⎜
⎟
⎜
= ⎜ 0
⎟ ⎜
⎟
⎟ ⎜
⎟
⎜
⎟ = ⎜
⎝
∗ ⎠ ⎜
∗ ⎟
⎜
⎟
⎝
0 ... 0
0 ⎠
0 ··· 0
Null(f − cidv)
f n
f n ≡ 0
Null(f − cidV ) d

f : V → V dim V = n
pf = (t − c1) r1 ···(t − ck) rk
g := f − c1idV
d := min{l ∈ N|Null g l = Null g l+1 }
ci
min{l ∈ N|Null g l = Null g l+1 } =min{l ∈ N|Bild g l = Bild g l+1 }
g| Bild g d : Bild g d → Bild g d
g| Null g d : Null g d → Null g d
i ∈ N
V = Null g d ⊕ Bild g d
N d ≡ 0
pf = �k ri
i=1 (t − ci)
f =
⎛
f(Vi) ⊂ Vi
V = V1 ⊕ ...⊕ Vk
(g| Null g d) d ≡ 0
Bild g d+i = Bild g d
Null g d+i = Null g d
g =
� N 0
0 C
�
dim Null gd
pg|Null gd = t
m g|Null g d
= td
dim Null g d = r1 ≥ d
+ Nj 0
⎝ cj · id dj Null gj 0 cjid dj Bild gj Vi := Null(f − ci · idV ) di
+ Cj
⎞
⎠

d ∈ N
f = fD + fN
f = fD + fN
a) fD
b) ∃m ∈ N : f m N ≡ 0
c) fD ◦ fN = fN ◦ fD
∀v ∈ V : g l+1 (v) = g l (g(v) ) ∈ Bild g
����
∈V
l
∀v ∈ Null g l : g l+1 (v) = g(g l (v)) = g(0) = 0
Bild g l+1 ⊂ Bild g l
Null g l+1 ⊃ Null g l
Bild g l+1 = Bild g l
Bild g l+1 ⊂Bild g l
⇐⇒ dim Bild g l+1 =dimBild g l
dim Null g l+i +dim Bild g l+i =dim V
⇐⇒ dim Null g l+1 =dimNull g l
Null g l+1 ⊃Null g l
⇐⇒ Null g l+1 = Null g l
v ∈ Bild g d w ∈ V v = g d (w)
g(v) =g(g d (w)) = g d+1 (w) ∈ Bild g d+1 = Bild g d
g| Bild g d : Bild g d → Bild g d
∀v ∈ Bild g d+1 ∃w ∈ V : g d+1 (w) = v
g(Bild g d )=Bild g d
g( g d (w)
� �� �
∈Bild gd ) = v

i =1:
i → i +1:
v ∈ Null g d
g d (g(v)) = g(g d (v))
= g(0) = 0
g(v) ∈ Null g d
g| Null g d : Null g d → Null g d
v ∈ Null g d ⇐⇒ g d (v) =0
� �d g|Null gd ≡ 0
g i g d (w) = g i+d (w) ∈ Bild g d+i
g d+i (w) = g i (g d (w)) ∈ g i (Bild g d )
∀i ∈ N : g i (Bild g d )=Bild g d+i
Bild g d+i+1 = g i g(Bild g d )
i=1 i d
= g (Bild g )
= Bild g d+i = Bild g d
Bild g d+i = Bild g d
⇐⇒ dim Bild g d+i =dimBild g d
dim Bild g d+i +dim Null g d+i =dim V
⇐⇒ dim Null g d+i =dimNull g d
Null g d+i ⊃Null g d
⇐⇒ Null g d+i = Null g d
v ∈ Null g d ∩ Bild g d
g d (v) = 0
∃w ∈ V : v = g d (w)

g d | Null g d ≡ 0 N d ≡ 0
g d | Bild g d
g 2d (w) =g d (g d (w)) = g d (v) =0
w ∈ Null g 2d = Null g d
v = g d (w) = 0
Null g d ∩ Bild g d = {0}
dim Bild g d +dimNull g d =dimV
V = Bild g d ⊕ Null g d
g(Null g d ) ⊂ Null g d
g(Bild g d ) = Bild g d
g =
� N 0
0 C
p(g d | Null gd) = det(t · idNull gd − g| d
Null gd) m(g| Null g d)=t l
�
=
⎛
⎜
det⎝
t −N d
⎞
⎟
⎠
0 t
dim Null gd
= t
g d | Null g d ≡ 0
m(g| Null g d)=t l
1 ≤ l ≤ dim Null g d
1 ≤ l ≤ d

l

�
N 0
f = gj + cjidV =
0 C
⎛
=
+ Nj 0
�
+ cjidV
⎝ cj · id dj Null gj 0 cjid dj Bild gj g(Vi) = (f − ci · id)(Vi) ⊂ Vi
f(Vi) = (f − ci · id + ci · id)(Vi) ⊂ Vi
det(f| Null g d j
j
+ Cj
− ci · id| dj )
Null gj = det(Nj +(cj − ci)id| Null g d j
j
= (cj − ci) rj
cj�=ci
�= 0
(f − ci · id)| Null g d j
j
Null (f − ciidV ) di ∩ Null (f − cjidV ) dj = {0}
Vi ∩ Vj = {0}
f(Vi) ⊂ Vi
k�
dim Vi =
i=1
gi| Null g d i
i
k�
ri = n
i=1
V = V1 ⊕···⊕Vk
)
i =1,...,k
= f|Vi − ciidVi = Ni
0 ≡ g di
i | Null g di = g
i
di
i |Vi
f|Vi = f|Vi − ciidVi + ciidVi = g + ciidVi
⎞
⎠

f =
=
D + N
⎛
c1idV1
⎜
⎝
0
⎞ ⎛
⎟ ⎜
⎠ + ⎝
0 ckidVk
DN, ND
⎛
⎜
D · N = ⎝
Ni
c1N1
0
0 ckNk
ci
⎞ ⎛
⎟ ⎜
⎠ = ⎝
⇒ di
⇒ Vi := Null (f|Vi − ciidVi) di
⇒ Ni := f|Vi − ciidVi
N1c1
N1
0
0 Nk
0
0 Nkck
⎞
⎟
⎠
⎞
⎟
⎠ = N · D
g : V → V d ≥ g d ≡ 0
d =min{l ∈ N : g l ≡ 0}
⎛
⎜
g = ⎜
⎝
Jd
Jd
J1
0 J1
0
⎞
⎟
⎠

Null g i
Jk :=
g d ≡ 0 Null g d = V
i>1 0 �= v ∈ Wi
Null g|Wi
g(v) ∈ Wi−1
⎛
0 1 0 0
⎞
⎜
⎝
0
1
⎟ ∈ M(k × k, C)
⎠
0 0
J l k �= 0 l1
= {0} g|Wi
w d 1,...,w d sd
Wd

g : Wd → Wd−1
Wd−1
Wd−1
g(w d 1),...,g(w d sd )
g(w d 1),...,g(w d sd ),wd−1 1 ,...,w d−1
sd−1
g d−1 (w d 1),...,g d−1 (w d sd ), ..., g(w2 1),...,g(w 2 s2 )
w 1 1,...,w 1 s1
W1
w d 1,g(w d 1),...,g d−1 (w d 1),...,w d sd ,g(wd sd ),...,gd−1 (w d sd )
w d−1
1
,g(w d−1
1
w 1 1,...,w 1 s1
W1
),...,g d−2 (w d−1
1 ),...,w d−1
sd−1 ,g(wd−1 sd−1 ),...,gd−1 (w d sd )
∀l

⎛
pf = det⎝
f − c · idV =
(f − c · idV ) 2 =
⎛
⎝
⎛
⎝
t − c
0
0
t − c
0
−1
0 0 t − c
0 0 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
mf = (t − c) 2
⎞
⎠
⎞
⎠
⎞
⎠ =(t− c) 3

x ∈ U s ∈ R n
f : R n → R
U ⊂ R n f : U ⊂ R n → R
h :[0, 1] → R n ,t↦→ x + ts
h([0, 1]) u ∈ [0, 1]
⎛
f(x + s) =
k� 1
⎝
j!
n� ∂
∂xij
... ∂
⎞
f(x)si1 ...sij
⎠
∂xi1
j =1
j=0
+
1
(k +1)!
g (k +1)
1 ≤ j ≤ k +1
djg (t) =
dtj i1,...,ij=1
n�
∂
∂xik+1
i1,...,ik+1=1
g :[0, 1] → R,t↦→ f(x + ts)
[0, 1]
g=f◦h
→ R
h ↘
U ⊂ R
↗ f
n
n�
i1,...,ij=1
... ∂
f(x + us)si1
∂xi1
...sik+1
∂
...
∂xij
∂
f(x + ts)si1
∂xi1
...sij
dg
(t)
dt
= Df(x + ts) · s
=
n� ∂
f(x + ts)si
∂xi
i=1

j → j +1
dj+1g (t)
dtj+1 = d
⎛
⎝
dt
n�
⎛
=
n� ∂
⎝
∂xm
=
m=1
i1,...,ij=1
n�
∂
...
∂xk
∂
⎞
f(x + ts)si1 ...sij
⎠
∂xi1
n� ∂
∂xij
... ∂
⎞
f(x + ts)si1 ...sij
⎠
∂xi1
d
dt (xm + tsm)
i1,...,ij=1
∂
∂xij+1
i1,...,ij+1=1
u ∈ [0, 1]
j=0
∂
...
∂xij
∂
f(x + ts)si1
∂xi1
i1,...,ij=1
R
...sij sij+1
f(x + s) =
k� g
g(1) =
j=0
j (0)
j! + gk+1 (u)
(k +1)!
⎛
=
k� 1
⎝
j!
n� ∂
∂xij
... ∂
⎞
f(x)si1 ...sij
⎠
∂xi1
+
f : R n → R
n�
∂
∂xik+1
i1,...,ik+1=1
... ∂
f(x + us)si1
∂xi1
...sij+1
f(x + s) = f(x)+〈Df(x),s〉 + 1
〈s, As〉 + r(s)
2
r(s)
lim
s→0 � s �
= 0
n × n A = A T
aij = ∂ ∂
f(x)
∂xi ∂xj

f(x + s) =
n� ∂
f(x)+ f(x)si +
∂xi i=1
1
n� ∂ ∂
f(x)sisj
2 ∂xi ∂xj
i,j=1
+ 1
n�
�
∂ ∂
f(x + us) −
2 ∂xi ∂xj
i,j=1
∂
�
∂
f(x) sisj
∂xi ∂xj
� �� �
=:r(s)
= f(x)+〈Df(x),s〉 + 1
〈s, As〉 + r(s)
2
A = A T
aij = ∂ ∂
f(x) =
∂xi ∂xj
∂ ∂
f(x) =aji
∂xj ∂xi
0 ≤
|r(s)|
lim
s→0 � s �2 ≤
n�
�
�
lim �
∂
s→0 �∂xi
i,j=1
= 0
sisj
�s�2 ≤ 1
∂
f(x + us) −
∂xj
∂
�
∂ �
f(x) �
∂xi ∂xj
�
f : U ⊂ R n → R x ∈ U
ε>0
g ′ i (0) = 0
Df(x) =0
gi :[0, 1] → R,t↦→ f(x + tei)
g ′ (t) =〈Df(x + tei),ei〉 = ∂
f(x + tei)
∂xi
gi

Df(x) =0
Df(x) =0
∀1 ≤ i ≤ n :
∂
f(x) =g
∂xi
′ i(0) = 0
f : U ⊂ R n x ∈ U
f(x + s) =f(x)+ 1
〈s, As〉 + r(s)
2
A = A T y1,...,yn
cn > 0 s = � n
i=1 biyi �= 0
〈s, As〉 =
=
n�
n�
i=1 j=1
n�
i=1 j=1
bibj 〈yi,Ayj〉
n�
bibj 〈yi,cjyj〉 =
n�
δ>0 � s �< δ
≥ cn b
i=1
2 i = cn 〈s, s〉 > 0
|r(s)| ≤ cn
4
� s �2
n�
i=1
b 2 i ci
f(x + s) ≥ f(x)+ cn
2 � s �2 − cn
=
� s �2
4
f(x)+ cn
� s �2
4
> f(x)

cn
〈s, As〉 =
=
≤
n�
n�
i=1 j=1
n�
i=1 j=1
n�
i=1
bibj 〈yi,Ayj〉
n�
bibjcj 〈yi,yj〉 =
b 2 i cn = cn 〈s, s〉
n�
i=1
b 2 i ci
f(x + s) ≤ f(x) − cn
2 � s �2 + cn
=
� s �2
4
f(x) − cn
� s �2
4
< f(x)
� v �< δ c1 > 0
f(x + v) >f(x)
� w �< δ cn < 0
f(x + w)

x 2 + y 4 ≥ 0
Df2(0, 0) =
A =
Df1(0, 0) =
A =
Df(0, 0) =
A =
� � � �
−2 · 0 0
=
−2 · 0 0
� �
−2 0
0 −2
� � � �
2 · 0 0
=
−2 · 0 0
� �
2 0
0 −2
� �
0
0
�
2 0
�
0 0
x 2 + y 4 =0 ⇐⇒ (x, y) =(0, 0)
f(x, y) =x 2 ≥ 0 y ∈ R f(0,y)=0
y0
f(0,y)=y 3 < 0
f(0,y)=y 3 > 0

f : X ⊂ C → C
�·�∞: W → [0, ∞),f ↦→ sup |f(x)|
x∈X
f K>0
�·�∞: W → [0, ∞)
sup
x∈X
∀x ∈ X : |f(x)| ≤K
|f(x)| + |g(x)| ≤ sup |f(x)| +sup|g(x)|
x∈X
x∈X
(|f(x)| + |g(x)|) ≤ sup
x∈X
� f �∞ = sup |f(x)| ≥0
x∈X
� cf �∞ = sup
x∈X
|f(x)| +sup
x∈X
|cf(x)| = |c| sup |f(x)|
x∈X
|g(x)|
= |c| �f �∞
� f + g �∞ = sup |f(x)+g(x)| ≤sup |f(x)| + |g(x)|
x∈X
x∈X
≤ sup |f(x)| +sup|g(x)|
=� f �∞ + � g �∞
x∈X
x∈X
f : X → C ⇐⇒
� f �∞= sup|f(x)|
=0
x∈X
⇐⇒ ∀x ∈ X : |f(x)| =0
⇐⇒ ∀x ∈ X : f(x) =0
⇐⇒ f ≡ 0
fn : X → C
∀x ∈ X : lim
n→∞ |fn(x) − f(x)| =0

C ⇐⇒
f : X → C
fn : X → C f : X →
lim
n→∞ � fn − f �∞,X= 0
fn : X → C fn
x ∈ X n ∈ N x ∈ X
|fn(x) − f(x)| < ε
3
fn δ>0 |x − y|

� b
lim
n→∞
a
fn(x)dx =
� b
a
fdx
[a, b] → R fn f f ′ n
fn
f ′ n
limn→∞ f ′ n
d
dx lim
n→∞ fn(x)
d
= lim
n→∞ dx fn(x)
f ′ n
fn(x) =fn(a)+
� x
f(x) = lim
n→∞ fn(x)
a
= f(a) + lim
n→∞
� x
f(x) =f(a)+
a
f ′ n(t)dt
� x
a
f ′ n(t)dt
lim
n→∞ f ′ n(t)dt
limn→∞ f ′ n
f ′ n limn→∞ f ′ n limn→∞ f ′ n
f ′ = lim
n→∞ f ′ n(x)
fn :

Ui0
limn→∞ yn = y
Ui
∃i0 : c ∈ Ui0
limn→∞ yn = c
yj ∈ Uij
(yk)k
{yn : n ∈ N}∪{y}
{yn : n ∈ N}∪{y} ⊂ �
i∈I
∃ε >0: B(c, ε) ⊂ Ui0
Ui
∃k0 ∀k ≥ k0 : yk ∈ B(c, ε) ⊂ Ui0
1 ≤ j ≤ k0
{yn : n ∈ N}∪{y} ⊂Ui0 ∪
[a, b] ⊂ R,U ⊂ R m
k0 �
j=1
Uij
f :[a, b] × U → R, (x, y) ↦→ f(x, y)
c := limk→∞ yk ∈ U
f(·,yk) :[a, b] → R,x↦→ f(x, yk)
f(·,c):[a, b] → R,x↦→ f(x, c)
Q := {yk : k ∈ N}∪{c}
[a, b] × Q ⊂ R 1+n

f :[a, b] × Q → R, (x, y) ↦→ f(x, y)
∀ε >0 ∃δ >0 ∀(x, y), (x ′ ,y ′ ) ∈ [a, b] × Q :
� (x, y) − (x ′ ,y ′ ) �< δ⇒|f(x, y) − f(x ′ ,y ′ )|

I,J ⊂ R
f : I × J → R, (x, y) ↦→ f(x, y)
c, yk ∈ J limk→∞ yk = c ∀k ∈ N : yk �= c
f(x, yk) − f(x, c)
yk − c
I × J
∀k ∈ N ∃sk
limk→∞ yk = c
f(x, yk) − f(x, c)
: I → R
yk − c
∂f
(x, c) :I → R
∂y
∂f
(x, y) :I → R
∂y
∂f
(x, c)
∂y
∀ε >0 ∃δ >0 ∀y, y ′ ∈ J :
|y − y ′ �
�
|

�
g : J → R,y ↦→ f(x, y)dx
I
�
dg
(y) =
dy I
∂f
(x, y)dx
∂y
c, yk ∈ J ∀k ∈ N : yk �= c limk→∞ yk = c
g ′ g(yk) − g(c)
(c) = lim
n→∞ yk − c
�
f(x, yk) − f(x, c)
= lim
dx
n→∞
I yk − c
�
∂f
=
(x, c)dx
∂y
∀c ∈ J g ′ (c)
∂f
∂y
I × J g ′ : J → R
� d
c
� d
c
I
f :[a, b] × [c, d] → R
F :[c, d] → R,y ↦→
F (y)dy =
� d
�� b
�
f(x, y)dx dy =
a
g :[c, d] → R,y ↦→
c
� b
a
[a, b] ⊂ R, [c, d] ⊂ R
f(x, y)dx
�� b
�
f(x, y)dx dy
a
� b
a
� b �� y
a
�� d
�
f(x, y)dy dx
c
c
�
f(x, t)dt dx

g(c) = 0
g ′ (y) =
=
=
=
� b
a
� b
a
� d
c
� d
c
� b
a
�� y �
∂
f(x, t)dt dx
∂y c
f(x, y)dx
�� b
�
f(x, y)dx dy
a
g ′ (y)dy = g(d)
�� d
�
f(x, y)dy dx
c

fn : A → C
� ∞
n=0 � fn �∞< ∞
� k
n=0 fn, � k
n=0 |fn| f,g : A → C
� ∞
n=0 � fn �< ∞ ∀ε >0 ∃N ∀k ≥ N
�
�
�
�
�
∞�
n=k+1
x ∈ B(a, r)
d
dx
�
�
�
fn(x) �
� ≤
∞�
n=k+1
∞�
n=k+1
� fn �∞< ε
|fn(x)| ≤
∞�
cn(x − a) n
n=0
x ∈ B(a, |x1 − a|)
0

q := < 1
|x1 − a|
|cn(x − a) n | = |cn(x1 − a) n � �
�
| �
x − a �
�
�x1
− a�
≤ M rn
= Mqn
|x1 − a| n
� cn(x − a) n �∞,B(a,r) ≤ Mq n
∞�
n=0
�
� N� �
� cn(x − a)
�
n=k
n
�
�
�
�
�
∞,B(a,r)
n=0
Mq n = M
1 − q
≤
<
N�
n=k
n
�cn(x − a) n � ∞,B(a,r)
∞�
Mq n

∞�
n=1
�
� N� �
� ncn(x − a)
�
n=k
n−1
�
�
�
�
�
∞,B(a,r)
n=1
� N
n=1 cn(x − a) n
� N
n=1 ncn(x − a) n−1
nMq n−1
≤
<
N� �
n=k
�ncn(x − a) n−1� � ∞,B(a,r)
∞�
nMq n−1

g ′ (x) =f(x, g(x))
y ′ = f(x, y)
U ⊂ R 2
f : U → R, (x, y) ↦→ f(x, y)
y ′ = f(x, y)
g : I ⊂ R → R
⇐⇒
a) Graph(g) ={(x, g(x)) ∈ I × R} ⊂U
b) ∀x ∈ I : g ′ (x) =f(x, g(x))
R
f(x, y) =f(x)
g ′ (x) =f(x)
g(x) =c +
g ′ (x) = d
�
c +
dx
G ⊂ R × R n
� x
� x
x0
x0
f(t)dt
�
f(t)dt = f(x)
f : G ⊂ R × R n → R n , (x, y) ↦→ f(x, y)
y ′ 1 = f1(x, y)
y ′ n = fn(x, y)

g : I ⊂ R → R n
⇐⇒
a) Graph(g) ={(x, y) ∈ I × R n : y = g(x)} ⊂U
b) ∀x ∈ I : g ′ (x) =f(x, g(x))
U ⊂ R × R n
f : U ⊂ R × R n → R n
L ≥ 0 ⇐⇒
(x, y), (x, z) ∈ U
� f(x, y) − f(x, z) �≤ L � y − z �
f : U ⊂ R × R n → R n ⇐⇒
∀(a, b) ∈ U ∃ V (a, b) ⊂ U : f : V → R n
g(x) =c +
� x
a
g(x) = g(a)+
= c +
∀(a, c) ∈ U ∃δ >0 ∃
� x
a
f(t, g(t))dt
� x
a
g ′ (t)dt
f(t, g(t))dt
U ⊂ R × R n f : U → R n
h :[a − δ, a + δ] → R n
B((a, c), 2r) ⊂ U ∀(x, y), (x, z) ∈ B((a, c), 2r)
y ′ = f(x, y) h(a) =c
� f(x, y) − f(x, z) �≤ L � y − x �

u ∈ BR(a, r
2 ) v ∈ BRn(c, r
2 )
� (u, v) − (a, c) �R×Rn = � (u − a, v − c) �R×Rn k =0
k → k +1
BR
�
a, r
2
≤ � (u − a, 0) �R×Rn + � (0,v− c) �R×Rn = � u − a �R + � v − c �Rn ≤ r r
+ = r0:� f(x, y) �∞,BR(a, r
2 )×BRn r (c, 2 )≤ M
�
r r
�
δ := min ,
3 2M
�
I := [a − δ, a + δ] ⊂ B a, r
�
2
h : I → R n ,x↦→ c +
� x
a
h0 : I → R n ,x↦→ c
∀k ∈ N0 : hk+1 : I → R n ,x↦→ c +
∀x ∈ I : hk(x) ∈ BR n
f(t, h(t))dt
� x
a
hk(x)
�
c, r
�
2
� h0(x) − c �=� c − c �= 0≤ r
2
f(t, hk(t))dt
� hk+1(x) − c � Def
=
≤
��
� x
�
�
�
� f(t, hk(t))dt�
�
a
��
� x
�
�
�
� � f(t, hk(t)) � dt�
�
a
≤ M|x − a| ≤Mδ
≤ M r r
≤
2M 2

hn
n → n +1: hn f(t, hn(t))
k =0
hn+1
k → k +1: x>a t ≥ a
� x
a
� x
a
|t − a| k dt =
x

hn
�
�
� ∞�
�
�
�
� (hk − hk−1)(x) �
�
� ≤
k=1
c +
≤
x ∈ I L �= 0
∞�
� (hk − hk−1)(x) �
k=1
∞�
k=1
≤ M
L
k−1 |x − a|k
ML
k!
∞� Lkδk k!
k=1
≤ M
L exp(Lδ)
∞�
(hk − hk−1)
k=1
h := lim
k→∞ hk = c +
∞�
(hk − hk−1) :I → R n
k=1
� f(x, hk(x)) − f(x, h(x)) � ≤ L � h(x) − hk(x) �
lim
k→∞ � f(·,hk(·)) − f(·,h(·)) �∞ ≤ L lim
k→∞ � h(·) − hk(·) �∞= 0
f(x, hk(x)) f(x, h(x)) f(x, hk(x))
h(x) = lim
k→∞ hk+1(x) = c + lim
= c +
= c +
h : I → R n ,x↦→ c +
k→∞
� x
� x
a
lim
a k→∞
� x
� x
a
a
f(t, hk(t))dt
f(t, hk(t))dt
f(t, h(t))dt
f(t, h(t))dt
y ′ = f(x, y) h(a) =c
f(t, h(t))

c + � x
f(t, h(t))dt
a
�
d
c +
dx
� x
a
�
f(t, h(t))
= f(x, h(x))
h ′ (x) = f(x, h(x))
h(a) = c
U ⊂ R × R n f : U → R n
h, g : I → R n y ′ = f(x, y)
I ⊂ R
h(a) =g(a) a
� h(x) − g(x) �
∀x ∈ I : h(x) =g(x)
h ′ (x) =f(x, h(x)) g ′ (x) =f(x, g(x))
h(x) = h(a)+
= h(a)+
g(x) = g(a)+
� x
a
� x
a
� x
a
h ′ (t)dt
f(t, h(t))dt
f(t, g(t))dt
δ>0 g ≡ h B(a, δ)
0

|x − a| ≤δ
0 0:[y, y + δ] ⊂ I
M(δ)(Lδ − 1)
� �� �
0 x ∈ B(y, δ)
g(x) =h(x)

x0 ∈ I,c ∈ R
g(x0) =c
g(x)
a, b : I ⊂ R → R
y ′ = a(x)y + b(x)
�� x
g : I → R,x↦→ c exp
g(x0) = c exp
g ′ (x) = c exp
y ′ = a(x)y
�� x0
x0
�� x
= g(x)a(x)
x0
x0
�
a(t)dt
�
a(t)dt = c · e 0 = c
� � x
d
a(t)dt a(t)dt
dx
x0
g : I × R → R, (x, y) ↦→ a(x)y + b(x)
a(x) [x0 − ε, x0 + ε] K>0
∂g
(x, y) =a(x)
∂y
g(x, z) − g(x, y) = ∂g
(x, s)(z − y) =a(x)(z − y)
∂y
|g(x, z) − g(x, y)| =
�
�
�
�
∂g
�
� (x, s)(z − y) �
∂y �
= |a(x)||z − y| ≤K|z − y|

h(x0) =c
y ′ = a(x)y + b(x)
� � x
b(t)
h(x) =g(x) c +
g(t) dt
�
g : I → R y ′ = a(x)y g(x0) =1
u : I → R
u : I → R
x0
h(x) =g(x)u(x)
h ′ = g ′ u + gu ′ = agu + gu ′
ah + b = agu + b
gu ′ = b
∀t ∈ I : g(t) =exp
� x
x0
a(t)dt �= 0
c = h(x0) =g(x0) u(x0) =u(x0)
� �� �
=1
u ′ (t) = b(t)
u(x) =
g(t)
� x
b(t)
c +
g(t) dt
x0
h ′ = ah + b
|f(x, z) − f(x, y)| ≤ sup
x∈B(x0,ε)
|a(x)||z − y| ≤K|z − y|
� � x
b(t)
h(x) =g(x) c +
g(t) dt
�
x0

h ′ (x) = g ′ � � x
b(t)
(x) c +
x0 g(t) dt
�
+ g(x) b(x)
=
g(x)
� � x
b(t)
a(x)g(x) c +
x0 g(t) dt
�
+ b(x)
= a(x)h(x)+b(x)
h(x0) = g(x0)(c +0)=c

[x0 − δ, x0 + δ]
LH
x0 ∈ I,c ∈ K n
A : I ⊂ R → M(n × n, K)
b : I ⊂ R → K n
g : I ⊂ R → K n
g(x0) =c
y ′ = A(x)y + b(x)
[x0 − δ, x0 + δ] ⊂ I. � A(x) �
∃K >0 ∀x ∈ [x0 − δ, x0 + δ] :� A(x) �≤ K
� f(x, y) − f(x, z) �=� A(x)(y − z) �≤ K � y − z �
LH
g1,...,gn
∃x0 ∈ I : g1(x0),...,gn(x0)
∀x0 ∈ I : g1(x0),...,gn(x0) ∈ Kn y ′ = A(x)y
∃x0 ∈ I : det(g1(x0),...,gn(x0)) �= 0
∀x0 ∈ I : det(g1(x0),...,gn(x0)) �= 0
dim LH = n
0 ∈ LH
g, h d ∈ K
0 ′ =0=A(x)0
(g + dh) ′ = g ′ + dh ′
= Ag + dAh
= A(g + dh)
g : I ⊂ R → K n
K

⇒ x0 ∈ I x0 �
∈ I
n
⇒
i=1 cigi ≡ 0
n�
cigi(x0) =0
i=1
g1(x0),...,gn(x0) 1 ≤ i ≤ n
ai =0
g1,...,gn
⇒ x0 g1(x0),...,gn(x0)
gj
⇔ ⇔
dim LH ≥ n
dim LH ≤ n
n�
i=1,i�=j
g :=
digi(x0) =gj(x0)
n�
i=1,i�=j
digi
y ′ = A(x)y g(x0) =gj(x0)
gj(x0) =g(x0)
gj ≡ g =
n�
i=1,i�=j
digi
det(v1,...,vn) �= 0 ⇐⇒ v1,...,vn
gi(x0) =ei
1 ≤ i ≤ n
g1(x0),...,gn+1(x0) ∈ K n
dim K n = n
g1,...,gn
y ′ = A(x)y + b(x) LI
LI = h + LH

⊂ g y ′ = Ay + b u := g − h
u ′ = g ′ − h ′ =(Ag + b) − (Ah + b) =Au
g = h + u ∈ h + LH
LI ⊂ h + LH
⊃ g = h + u u y ′ = Ay
A(x)y
h(x0) =c
g ′ = h ′ + u ′ =(Ah + b)+Au = Ag + b
g ∈ LI
h + LH ⊂ LI
G =(g1,...,gn) LH y ′ =
G ′ =(g ′ 1,...,g ′ n)=(Ag1,...,Agn) =AG
h : I → K n
u : I → K n ,x↦→
y ′ = A(x)y + b(x)
h(x) =G(x)u(x)
� x
G(t)
x0
−1 b(t)dt + G −1 (x0)c
Ah + b = h ′
G −1
= Gu ′ + G ′ u
G ′ =AG
= Gu ′ + AGu
h=Gu
= Gu ′ + Ah
h ′ = Ah + b ⇐⇒ u ′ = G −1 b

sin wx cos wx
c = h(x0) =G(x0)u(x0)
u(x) = u(x0)+
� x
= G −1 (x0)c +
u ′ (t)dt
x0
� x
x0
G(t) −1 b(t)dt
h ′
= Gu ′ + Ah
= GG −1 b + Ah
h(x0) = G(x0)u(x0) =G(x0)G −1 (x0)c = c
LH = Lin
y ′ 1 = −wy2
y ′ 2 = wy1
�� � � ��
cos wx − sin wx
,
sin wx cos wx
y ′′
1 = −wy ′ 2 = −w 2 y1
y ′′
2 = wy ′ 1 = −w 2 y2
sin ′′ wx = −w 2 sin wx
cos ′′ wx = −w 2 cos wx
g1 =
g2 =
� �
cos wx
sin wx
� �
− sin wx
cos wx

�
cos wx − sin wx
det
sin wx cos wx
y ′ 1 = −y2
y ′ 2 = y1 + x
�
=1�= 0
g1,g2
h(x0) =c
h(x) =
� ���
cos x − sin x cos x0
sin x cos x − sin x0
� � � �
cos x − sin x
+R + R
sin x cos x
sin x0
cos x0
� y1
y2
G −1 (t)b(t) =
� ′
=
G(x) =
G −1 (x) =
� 0 −1
1 0
� cos t sin t
− sin t cos t
�� y1
y2
�
+
�
c +
� �
0
x
�
cos x
�
− sin x
sin x
�
cos x
cos x
sin x
�
− sin x cos x
�� �
0
=
t
� x
x0
� �
t · sin t
t · cos t
� � �
t sin t
dt
t cos t

f : G ⊂ R × R n → R, (x, y) → f(x, y)
y (n) = f(x, y, y ′ ,...,y (n−1) )
g : I ⊂ R → R
{(x, y0,...,yn−1) ∈ I × R n : yi = g (i) (x), 0 ≤ i ≤ n − 1} ⊂G
g (n) (x) =f(x, g(x),g ′ (x),...,g (n−1) (x))
y (n) = f(x, y, y ′ ,...,y (n−1) )
y ′ 0 = y1
y ′ n−2 = yn−1
y ′ n−1 = f(x, y0,...,yn−1)
⇒ h(x) y (n) = f(x, y, y (1) ,...,y (n−1) )
⎛
⎜
Y = ⎜
⎝
y ′ 0 = y1
y ′ n−2 = yn−1
y ′ n−1 = f(x, y0,...,yn−1)
y0
yn−1
⎞
⎟
⎠
⎛
⎜
F (x, Y )= ⎜
⎝
y1
yn−1
f(x, Y )
⎞
⎟
⎠

⇐
H ′ (x) =
F (x, Y )
Y ′ = F (x, Y )
⎛
H : I → R n ⎜
,x↦→ ⎜
⎝
=
⎛
⎜
⎝
⎛
⎜
⎝
h(x)
h ′ (x)
h (n−1) (x)
h (1) (x)
h (2) (x)
h(x)
h ′ (x)
⎞′
⎛
⎟
⎠
⎜
= ⎜
⎝
h (n−1) (x)
⎞
⎟
⎠
h (1) (x)
h (2) (x)
h (n) (x)
⎞
⎟
⎠
f(x, h(x),h (1) (x),...,h (n−1) (x))
= F (x, H(x))
⎛
⎜
G = ⎝
g0
gn−1
y ′ 0 = y1
⎞
⎟
⎠ : I → R n
y ′ n−2 = yn−1
y ′ n−1 = f(x, y0,...,yn−1)
g := g0 : I → R
⎞
⎟
⎠

h0 ∈ LI
g1 = g ′ 0 = g ′
g2 = g ′ 1 = g ′′
gn−1 = g ′ n−2 = g (n−1)
g (n) = g ′ n−1 = f(x, g0,...,gn−1)
= f(x, g(x),g ′ (x),...,g (n−1) (x))
0 ≤ k ≤ n − 1
b, ak : I ⊂ R → K
y (n) + an−1(x)y (n−1) + ...+ a0(x)y = b(x)
b ≡ 0 g : I → K
LH
LI u : I → K b �≡ 0
LI = h0 + LH
g1,...,gn ∈ LH
(∀ ⇐⇒ ∃)x∈I ⇐⇒
⎛
⎜
W (x) =det⎝
g1(x) ... gn(x)
g (n−1)
⎞
⎟
⎠ �= 0
(x)
1 (x) ... g (n−1)
n
⇐⇒
⎛
g
⎜
⎝
g ′
g (n−1)
⎞
⎟ : I → Kn
⎠

y ′ 0 = y1
y ′ n−2 = yn−1
y ′ n−1 = −a0(x)y0 − ...− an−1(x)yn−1 + b(x)
y ′ ⎛
0
⎜
= ⎜ 1
⎜
⎝
−a0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ y + ⎜
⎠ ⎝
0
0
⎞
⎟
⎠
1 −an−1
b

(Ai)i∈I
I p
A, B ∈ I p
A, B ∈ I p
A ∩ B = ∅
�
Ai := �
i∈I
i∈I
Ai
R 2 R n
⇐⇒
(a1,b1] × (a2,b2] (b1 − a1)(b2 − a2)
n�
(a1,b1] × ...× (an,bn]
(bi − ai)
A =
(A) =
R p
∞�
i=1
∞�
i=1
Ai
Ai
(Ai)
i=1
:= {(a, b] :a, b ∈ R p }
:= {(a1,b1] × ...× (ap,bp] :ai,bi ∈ R}
A\B =
A ∩ B ∈ I p
m�
j=1
Cj
Cj ∈ I p

p → p +1
∀j :max 2 i=1 ai,j < min 2 i=1 bi,j
(a11,b11] × ...× (a1p,b1p] ∩ (a21,b21] × ...× (a2p,b2p]
i=1 ai1,
2
min
i=1 bi1] × ...× ( 2
max
i=1 aip,
2
min
i=1 bip]
= ( 2
max
∃j :max 2 i=1 ai,j ≥ min 2 i=1 bi,j
(a11,b11] × ...× (a1p,b1p] ∩ (a21,b21] × ...× (a2p,b2p] =∅
((a21,b21] × (a22,b22]) C
= (−∞,a21] × R +(b21, ∞) × R
+(a21,b21] × (−∞,a22]+(a21,b21] × (b22, ∞)
A\B = (a11,b11] × (a12,b12] \ (a21,b21] × (a22,b22]
= (a11,b11] × (a12,b12] ∩ ((a21,b21] × (a22,b22]) C
=
4�
Cj ∈ I 2
j=1
Cj
((a21,b21] × ...× (a2,p+1,b2,p+1]) C
= (a21,b21] × ...× (a2p,b2p] × (b2,p+1, ∞)
+(a21,b21] × ...× (a2p,b2p] × (−∞,a2,p+1]
+((a21,b21] × ...× (a2,p,b2,p]) C × (a2,p+1,b2,p+1]

A\B
= (a11,b11] × ...× (a1,p+1,b1,p+1]
∩ ((a21,b21] × ...× (a2,p+1,b2,p+1]) C
= (a11,b11] × ...× (a1,p+1,b1,p+1]
∩(a21,b21] × ...× (a2p,b2p] × (b2,p+1, ∞)
+(a11,b11] × ...× (a1,p+1,b1,p+1]
∩(a21,b21] × ...× (a2p,b2p] × (−∞,a2,p+1]
+(a11,b11] × ...× (a1,p+1,b1,p+1]
∩ ((a21,b21] × ...× (a2,p,b2,p]) C × (a2,p+1,b2,p+1]
p=1
= C1 + C2 +
((a11,b11] × ...× (a1p,b1p]\(a21,b21] × ...× (a2,p,b2,p])
× ((a1,p+1,b1,p+1] ∩ (a2,p+1,b2,p+1])
⎛ ⎞
= C1
����
∈Ip+1 + C2
����
∈Ip+1 =
Cj ∈ I p
n�
Cj
����
j=1
∈Ip+1 A, B1,...,Bn ∈ I p
n�
⎜
+ ⎝ Dj
����
j=3
∈Ip ⎟
⎠ × D0
����
∈I1 A\
n�
i=1
Bi =
m�
j=1
A ∈ I p
Cj

n =1
n → n +1
A\
n�
i=1
Bi
=
=
=
=
�
n�
A ∩
⎛
m�
⎝
i=1
Cj
j=1
B C i
�
∩ B C n+1
⎞
⎠ ∩ B C n+1
m�
Cj ∩ B C n+1
� �� �
j=1
m�
j=1 k=1
=Cj\Bn+1
lj�
Dkj
R ⊂ Pot(W ) ⇐⇒
∅∈R
A, B ∈ R A\B = A ∩ B C ∈ R
A, B ∈ R A ∪ B ∈ R
A, B ∈ R
A ∩ B ∈ R
Dkj ∈ I p
A ∩ B = A ∩ (A C ∪ B) =A ∩ (A ∩ B C ) C = A\(A\B) ∈ R
F p �
n�
�
�
�
:= (ai,bi] �
� ai,bi ∈ R p
�
∪{∅}
i=1
∅∈F p
A = �m k=1 Ak B = �n l=1 Bl ∈ F p
m�
� �
n�
��
m� nk �
A\B = Ak\ Bl = Cik
����
k=1
l=1
k=1 i=1
∈Ip ∈ F p

A ∪ B = A ∩ B C + B
=
m�
n�
Ak\B +
=
k=1
m�
∈ F p
tk�
k=1 s=1
Ck,s +
l=1
Bl
n�
l=1
Bl

A ⊂ B
m : R → [0, ∞] ⇐⇒
m(∅) = 0
�
n�
�
n�
m
= m(Ai)
i=1
Ai
i=1
m : R → [0, ∞] ⇐⇒
m(∅) = 0
∞�
�
∞�
�
∞�
Ai ∈ R m
= m(Ai)
i=1
A ⊂ B,m(A) < ∞
� ∞
i=1 Ai ∈ R
i=1
Ai
m(A) ≤ m(B)
m(B\A) =m(B) − m(A)
m
� n�
i=1
Ai
�
≤
n�
m(Ai)
i=1
∞�
�
∞�
m(Ai) ≤ m
i=1
i=1
i=1
A, B, Ai ∈ R i ∈ N
Ai
m(A ∪ B)+m(A ∩ B) =m(A)+m(B)
� ∞
i=1 Ai ∈ R
m
� ∞�
i=1
Ai
�
�
∞�
≤ m(Ai)
i=1

m(A) < ∞
m
� n�
i=1
m(B) = m(A + B ∩ A C )
= m(A)+m(B ∩ A C ) ≥ m(A)
� �� �
≥0
m(B\A) =m(B ∩ A C )=m(B) − m(A)
Ai
�
⎛ ⎛
n�
= m ⎝ ⎝Ai ∩
=
Ai∩ T i−1
j=1 Aj⊂Ai
≤
i=1
⎛
n�
m ⎝Ai ∩
i=1
n�
m(Ai)
i=1
�∞ i=1 Ai\ �n i=1 Ai ∈ R ∀n ∈ N
n�
�
n�
�
m(Ai) = m
i=1
≤ m
= m
= m
i=1
� n�
i=1
� n�
i=1
� ∞�
i=1
Ai
Ai
� ��
∞�
+ m
Ai +
Ai
limn→∞
�
∞�
i=n+1
∞�
�
∞�
m(Ai) ≤ m
i=1
i=1
Ai
i−1 �
j=1
i−1 �
j=1
A C j
A C j
⎞
⎠
�
n�
\
⎞⎞
⎠⎠
Ai
i=1
�
� �� �
i=1
Ai
Ai
�
�
∈R
m(A) =∞ m(B) =∞ m(A ∪ B) =∞
∞ = ∞

m(A),m(B) < ∞
m(A ∪ B) = m(A + B ∩ A C )
= m(A)+m(B ∩ A C )
= m(A)+m(B) − m(A ∩ B)
m(A ∩ B) ≤ m(A) < ∞ m(A ∩ B)
E ⇐⇒
m
� ∞�
i=1
Ai
Ei ↑ E ⇐⇒
(Ei)i∈N
�
⎛ ⎛
∞�
= m ⎝ ⎝Ai ∩
=
i)
≤
An ↑ � ∞
n=1 An
i=1
⎛
∞�
m ⎝Ai ∩
i=1
∞�
m(Ai)
i=1
(Ei)i∈N
E1 ⊂ E2 ⊂ ...
∞�
E =
i−1 �
j=1
A C j
⎞
i−1 �
A
j=1
C⎠ j
� �� �
=Ai\ Si−1 j=1 Aj∈R
i=1
Ei
E1 ⊃ E2 ⊃ ...
∞�
E =
A C n ↓
i=1
∞�
n=1
Ei
A C n
⎞⎞
⎠⎠
Ei ↓

An ↓ � ∞
n=1 An
An ↓ � ∞
n=1 An
En ↑ E
An ⊃ An+1
An ⊃ An+1
An ⊂ An+1
An ∩
An ∩
A C n ↑
� ∞�
n=1
∞�
n=1
An
A C n
� C
↓∅
∀w ∈ E C ∀n ∈ N : w ∈ E C n
A C n ⊂ A C n+1
� ∞�
n=1
⎛
∞�
�
∞�
⎝An ∩
n=1
n=1
w ∈ E C = � ∞
n=1 EC n
An
An
m : R → [0, ∞]
� C
A C n ⊃ A C n+1
⊃ An+1 ∩
� ⎞
C ∞�
⎠ =
n=1
An ∩
∀n ∈ N : w ∈ E C n
∞�
�
∞�
m(An) =m
n=1
lim
n→∞ m(An) =m
n=1
� ∞�
An
n=1
An
� ∞�
�
n=1
An ∈ R An ↑ � ∞
� ∞�
n=1
An
�
� C
An
� C
= ∅
n=1 An ∈ R

m(W ) < ∞
�
∞�
m
⇒
⇒
n=1
An
An ∈ R, m(A1) < ∞ An ↓ � ∞
n=1 An ∈ R
lim
n→∞ m(An) =m
� ∞�
n=1
An
∅ An ∈ R, m(An) < ∞ An ↓∅
lim
n→∞ m(An) =0
a) ⇐⇒ b) =⇒ c) ⇐⇒ d)
�
a) ⇐⇒ b) ⇐⇒ c) ⇐⇒ d
⇒ An−1 ⊂ An A0 := ∅
�
An−1⊂An
=
�
∞�
�
m An ∩ A C �
n−1
�
a)
=
∞�
n=1
n=1
= lim
k→∞ m
� n
i=1 Ai ⊂ � n+1
i=1 Ai
∞�
m(Ai) = lim
i=1
m( An ∩ A C n−1 ) = lim
� �� � k→∞
=An\An−1∈R
�
k�
�
n�
n→∞
i=1
�
∞�
b)
= m
An ↓
n=1
n=1 i=1
∞�
n=1
An
An ∩ A C n−1
m(Ai) = lim
n→∞ m
n�
Ai
� �
∞�
= m
A C n ↑
∞�
n=1
i=1
k�
m(An ∩ A C n−1)
n=1
= lim
k→∞ m(Ak)
� n�
i=1
Ai
�
� �� �
A C n
A1 ∩ A C ∞�
n ↑ (A1 ∩ An) C ∞�
= A1 ∩
n=1
n=1
Ai
∈R
�
A C n

⇒
⇒
lim
n→∞ m(An)
A1 = A1 ∩ An + A1 ∩ A C n
An⊂A1
= An + A1 ∩ A C n
A1 =
∞� ∞�
A1\ An +
n=1
n=1
An
A1=An+A1∩A C � n
= lim m(A1) − m
n→∞
� A1 ∩ A C�� n
A1∩A C
n ↑A1∩S ∞
= m(A1) − lim
n→∞ m � A1 ∩ A C n=1
�
n
AC
=
n
�
∞�
m(A1) − m A1 ∩ A
n=1
C �
n
=
�
∞�
m(A1) − m A1\
�
= m
� ∞�
n=1
An
�
∞�
�
c)
lim m(An) = m An = m(∅) =0
n→∞
n=1
⎛
∞�
�
∞�
⎝An ∩
n=1
An ⊂ An−1
n=1
An ↓
An
∞�
n=1
� ⎞
C �
∞�
⎠ =
An
m(A1) < ∞
n=1
An ∩
An
�
� �
∞�
∩
� ∞�
n=1
∀n ∈ N : m(An) < ∞
An
n=1
� C
An
An
n=1
↓∅
� C
= ∅

lim
n→∞ m(An)
= lim
n→∞ m
⎛ �
∞�
� ⎞
C
⎝An ∩ An ⎠ + lim
n→∞
n=1
m
�
�
∞�
� �
∞�
�
d)
= m(∅) +m An = m An
� �� �
n=1
n=1
=0
⇒ m(W ) < ∞
An ↑
∞�
n=1
An
A C n ↓
∞�
n=1
A C n
∞�
An ∩ An
n=1
lim
n→∞ m(An)
� � �� C
= lim m(W ) − m An n→∞
= m(W ) − lim
n→∞ m � A C� n
�
∞�
�
c)
= m(W ) − m
n=1
A C n
⎛�
∞�
= m(W ) − m ⎝
�
∞�
m(W )

lim
n→∞ m(An) =∞ �= 0=m(∅) =m
l : F p m�
→ [0, ∞), (a j ,b j m� p�
] ↦→
j=1
⎛
m�
l ⎝ (a j ,b j ⎞
] ⎠ =
j=1
=
m�
� ∞�
n=1
(b
j=1 i=1
j
i
p�
(b
j=1 i=1
j
i
− aj
i )
m�
l � (a j ,b j ] �
j=1
An
�
− aj
i )
l
A = � m
j=1 (aj ,b j ]= � n
k=1 (ck ,d k ] (a j ,b j ], (c k ,d k ] ∈ I p
m�
(a j ,b j ] =
j=1
=
=
=
m�
(a j ,b j ] ∩
j=1
m�
n�
j=1 k=1
k=1
n�
(c k ,d k ]
k=1
�
m,n
max
j,k=1 (aj ,c k ), m,n
min
j,k=1 (bj ,d k �
)
n�
(c k ,d k m�
] ∩ (a j ,b j ]
n�
(c k ,d k ]
k=1
j=1

⎛
m�
l ⎝ (a j ,b j ⎞
] ⎠ =
j=1
=
=
m�
l � (a j ,b j ] �
j=1
m�
n�
j=1 k=1
��
m,n
l max
j,k=1 (aj ,c k ), m,n
min
j,k=1 (bj ,d k ��
)
n�
l � (c k ,d k ] �
k=1
�
n�
= l (c k ,d k �
]
k=1
l A = �m j=1 Aj,B = �n k=1 Bk
⎛
⎞
l(A + B) =
m� n�
l ⎝ Aj + ⎠
=
j=1
m�
l(Aj)+
j=1
⎛
m�
= l ⎝
Aj
j=1
l I p
� ∞
j=1 (aj ,b j ]=(a, b] ∈ I p
� p
i=1
(x + bj
i
k=1
Bk
n�
l(Bk)
k=1
⎞ �
n�
⎠ + l
k=1
Bk
�
= l(A)+l(B)
Aj,Bk ∈ I p
− aj
i ) δj > 0 2x ≤ δj
�
� p� �
� (2x + b
�
j
i
i=1
− aj
i
) −
l((a j − δj,b j + δj]) =
p�
�
�
�
�
�
(b
i=1
j
i − aji
)
p�
(2δj + b j
i=1
< ε
2
p�
< ε
2 j
i
+ (b j
i=1
j
i
= l((a j ,b j ]) + ε
2 j
− aj
i )
− aj
i )

[a, b]
(a, b] ⊂ [a, b] ⊂
(a, b] =
[a, b] ⊂
∞�
(a j ,b j ]
j=1
∞�
(a j − δj,b j + δj)
j=1
(a j − δj,b j + δj)
n�
(a j − δj,b j + δj) ⊂
j=1
l F p
ε>0
l((a, b])
n�
(a j − δj,b j + δj]
j=1
� �� �
∈F p
⎛
i) n�
≤ l ⎝ (a j − δj,b j ⎞
+ δj] ⎠
iii)
≤
≤
≤
j=1
n�
l((a j − δj,b j + δj])
j=1
n�
j=1
�
l((a j ,b j ]) + ε
2j �
∞�
l((a j ,b j ]) + ε
j=1
l((a, b]) ≤
∞�
l((aj,bj])
j=1
∞�
�
∞�
�
l(ai,bi] ≤ l (ai,bi] = l((a, b])
i=1
i=1
∞�
�
∞�
�
l((ai,bi]) = l (ai,bi]
i=1
i=1

l F p
Ak = � nk
l=1 Ckl ∈ F p Ckl ∈ I p
� ∞
k=1 Ak ∈ F p
l(A) =
∞�
k=1
Ak =
Bj = Bj ∩
l(Bj) c)
=
=
=
m�
j=1
k=1
m�
Bj ∈ F p
j=1
∞�
k=1
k=1 l=1
Ak =
Bj ∈ I p
∞� nk �
(Bj ∩ Ckl)
k=1 l=1
∞� nk �
∞�
l(Bj ∩ Ckl) = l(Bj ∩ Ak)
l(Bj) c)
=
j=1
∞�
k=1 j=1
k=1
m�
l(Bj ∩ Ak)
⎛
⎞
∞� m�
∞�
�
l ⎝ (Bj ∩ Ak) ⎠ = l
∞�
l(Ak)
k=1
k=1
∞�
Ak ∩ Ak
k=1
�

S ⊂ Pot(W ) ⇐⇒
W ∈ S
A ∈ S AC ∈ S
Sj
(An)n∈N
a) Pot(W )
b) ∅∈S
c) (An)n∈N
� ∞
n=1 An ∈ S
∞�
An ∈ S
n=1
W ∈ Pot(W )
A ∈ Pot(W ) ⇒ A C ∈ Pot(W )
∞�
∀n ∈ N : An ∈ Pot(W ) ⇒ An ∈ Pot(W )
∞�
n=1
(Sj)j∈J
(Ai)i∈N
An =
n=1
∅ = W C ∈ S
�
∞�
�C ∈ S
n=1
A C n
S := �
j∈J
S = �
j∈J Sj
Sj
∀j ∈ J : Ai ⊂ Sj
W ∈ Sj
A C i ∈ Sj
�
i∈N
Ai ∈ Sj
∀j ∈ J

R p
W ∈ �
j∈J
A C i ∈ �
j∈J
�
Ai ∈ �
i∈N
j∈J
Sj
Sj
Sj
E ⊂ Pot(W ) (Sj)j∈J
Sj
S(E) = �
j∈J
�
Sj �= ∅
j∈J
S(E) =Sk
Sj
B p := S( R p )
B p = S( R p )
B p = S(I p )
S(F p )=S(I p )
Sk
k ∈ J

⊂
⊃
A ∈ S ⇐⇒ A C ∈ S
⇐⇒ A C
S( R p )=S( R p )
rx
y ∈ U ∩ Q C
εx
Q R
�
(a, b] =
∞�
�
a, b + 1
�
n
� �� �
n=1
S(I p ) ⊂ S( R p )
∀x ∈ U∃rx : B(x, rx) ⊂ U
(x − εx,x+ εx) ⊂ B(x, rx) ⊂ U
�
x∈Q∩U
(x − εx,x+ εx) ⊂ U
∃εy > 0: (y − εy,y+ εy) ⊂ U
∃x ∈ Q ∩ U : � x − y �∞< εy
2
�
y ∈ x − εy
�
εy εx
,x− ⊂ (x − εx,x+ εx)
2 2
∞�
x∈Q∩U n=1
�
x∈Q∩U
�
x∈Q∩U
(x − εx,x+ εx) ⊃ U
(x − εx,x+ εx) = U
�
x − εx,x+ εx − 1
�
n
⊃ U
S(I p ) ⊃ S( R)

⊃ F p ⊃ I p
S(F p ) ⊃ S(I p )
⊂ S(I p ) Ci ∈
I p
F p ⊂ S(I p )
⇐⇒
D ⊂ Pot(W )
W ∈ D
A ∈ D AC ∈ D
Ai ∈ D
(Ai)i∈N
(Dj)j∈J
S(F p ) ⊂ S(I p )
� ∞
i=1 Ai ∈ D
W ∈ Pot(W )
A ∈ Pot(W ) ⇒ A C ∈ Pot(W )
∞�
Ai ∈ Pot(W ) ⇒ Ai ∈ Pot(W )
D := �
j∈J
i=1
Dj
D := �
j∈J Dj
∀j ∈ J : Ai ⊂ Dj

Dj
Pot(W )
Dj
W ∈ Dj
A C i ∈ Dj
�
i∈N
Ai ∈ Dj
W ∈ �
j∈J
A C i ∈ �
j∈J
�
Ai ∈ �
i∈N
j∈J
Dj
Dj
Dj
E ⊂ Pot(W ) (Dj)j∈J
D(E) = �
j∈J
�
Dj �= ∅
j∈J
�
j∈J Dj
Dj
∀A, B ∈ M : A ∩ B ∈ M
D(E) ⊂ S(E)
∀j ∈ J
⇐⇒
Dj

k>i
⎛
S(E) ⇒ D(E) =S(E)
� ∞
i=1 Ai
⎝Ai ∩
i−1 �
j=1
∞�
i=1
D(E) ⊂ S(E)
D(E) ⊃ S(E)
B ∈ D(E)
DB
W ∈ DB
DB
A C j
∞�
i=1
Ai =
⎞ ⎛
⎠ ∩ ⎝Ak ∩
Ai =
∞�
Ai ∈ S
i=1
k−1 �
j=1
⎛
∞�
⎝Ai ∩
i=1
A C j
i−1 �
j=1
⎞
⎠ ⊂ Ai ∩ A C i = ∅
A C j
� �� �
∈D
D(E) ⊃ S(E)
D(E) = S(E)
D(E) =S(E)
⎞
⎠ ∈ D
DB := {Q ⊂ W : Q ∩ B ∈ D(E)}
W ∩ B = B ∈ D(E)
D(E)

A ∈ DB
Ai ∈ DB
C1 ∈ E
A ∈ DB
Ai ∈ DB
A C ∈ DB
DB
⇐⇒ A ∩ B ∈ D(E)
⇐⇒ (A ∩ B) C ∈ D(E)
⇐⇒ A C ∪ B C ∈ D(E)
DB
⇐⇒ A C ∩ B =(A C ∪ B C ) ∩ B ∈ DB
� ∞
i=1 Ai ∈ DB
DB
⇐⇒ Ai ∩ B ∈ D(E)
⇐⇒
DB
⇐⇒
∞�
�
∞�
(Ai ∩ B) =
i=1
∞�
i=1
Ai ∈ DB
⊂ B C1,C ∈ E
C1 ∩ C ∈ E
i=1
Ai
�
∩ B ∈ D(E)
∀C ∈ E : C ∩ C1 ∈ E ⊂ D(E)
DC i
⇒ ∀C ∈ E : C ∈ DC1
⇒ E ⊂ DC1
D(E) ⇒ D(E) ⊂ DC1
∀C ∈ E : D(E) ⊂ DC
⇐⇒ ∀C ∈ E ∀B ∈ D(E) : B ∈ DC
DC
⇒ ∀C∈E∀B∈D(E) : B ∩ C ∈ D(E)
DB
⇒ ∀C∈E∀B∈D(E) : C ∈ DB
⇐⇒ ∀B ∈ D(E) :E ⊂ DB
D(E)
=⇒ ∀B∈D(E) : D(E) ⊂ DB
⇐⇒ ∀A, B ∈ D(E) :A ⊂ DB
DB
⇒ ∀A, B ∈ D(E) :A ∩ B ∈ D(E)
⇒ D(E)
S(E) =D(E)

An ∈ R
m : S → [0, ∞]
m(∅) = 0
�
∞�
�
∞�
m
= m(Ai)
i=1
Ai
Q ∈ Pot(W )
i=1
m : R → [0, ∞]
n=1
m : S(R) → [0, ∞]
m ∗ �
∞�
∞�
: → [0, ∞],Q↦→ inf m(An) | An ∈ ,Q⊂
m ∗ (Q) =∞ (An)n∈N Q ⊂ � ∞
n=1 An
m ∗ (∅) =0
Q1 ⊂ Q2
m ∗
m ∗ (Q1) ≤ m ∗ (Q2)
� ∞�
n=1
Qn
�
∞�
≤ m ∗ (Qn)
n=1
n=1
An
�

m ∗
m(An) ≥
∅∈R ∅⊂ � ∞
n=1 ∅
ε>0
0 ≤ m ∗ ∞�
∞�
(∅) = m(∅) = 0=0
Q2
n=1
Q1 ⊂ Q2 ⊂
Q2
Q1
∞�
n=1
An
m ∗ (Q1) ≤ m ∗ (Q2)
n=1
m ∗ ∀n ∈ N ∃A n i
m ∗
∞�
i=1
� ∞�
n=1
m ∗
Qn ⊂
∞�
i=1
A n i
m(A n i ) ≤ m ∗ (Qn)+ ε
2 n
Qn
∞�
n=1
�
� ∞�
n=1
Qn ⊂
≤
≤
∞�
i,n=1
∞�
i,n=1
∞�
n=1
= ε +
Qn
A n i
m(A n i )
Q1
�
m ∗ (Qn)+ ε
∞�
m ∗ (Qn)
n=1
�
∞�
≤ m ∗ (Qn)
n=1
2 n
�

m ∗ Q ⊂ W
S ∗ := � A ⊂ |∀Q ⊂ W : m ∗ (Q) =m ∗ (Q ∩ A)+m ∗ � Q ∩ A C��
m ∗ (Q)
W ∈ S ∗
m ∗ (Q ∩ W )+m ∗ � Q ∩ W C� = m ∗ (Q)+m ∗ (∅)
A ∈ S ∗ A C ∈ S ∗
= m ∗ (Q)
m ∗ (Q ∩ (A C ) C )+m ∗ (Q ∩ A C ) = m ∗ (Q ∩ A C )+m ∗ (Q ∩ A)
A, B ∈ S ∗ A ∩ B,A ∪ B ∈ S ∗
m ∗ � Q ∩ (A ∩ B) C�
= m ∗ � Q ∩ (A C ∪ B C ) �
= m ∗ (Q)
B∈S ∗
= m ∗ � Q ∩ (A C ∪ B C ) ∩ B � + m ∗ � Q ∩ (A C ∪ B C ) ∩ B C�
= m ∗ � Q ∩ A C ∩ B � + m ∗ � Q ∩ B C�
B∈S ∗
= m ∗ (Q ∩ B)+m ∗ � Q ∩ B C�
A∈S ∗
= m ∗ (Q ∩ A ∩ B)+m ∗ � Q ∩ A C ∩ B � + m ∗ � Q ∩ B C�
= m ∗ (Q ∩ A ∩ B)+m ∗ � Q ∩ (A ∩ B) C�
A ∩ B ∈ S ∗
A ∪ B =(AC ∩ BC ) C A ∪ B ∈ S∗ Ai ∈ S∗ �n i=1 Ai ∈ S∗ m ∗
�
Q ∩
n�
i=1
Ai
�
=
n�
m ∗ (Q ∩ Ai)
i=1

n =2 A = Ai B = Aj
m ∗ (Q ∩ (Ai + Aj)) =
�
∗
m Q ∩ � A C i ∩ A C�C j
�
3.)
=
∗
m � Q ∩ Ai ∩ A C� ∗
j + m (Q ∩ Aj)
n → n +1 An+1 ∩ �n i=1 Ai = ∅
m ∗
� �
n�
Q ∩ An+1 +
��
m ∗ (Q)
Ai ∈ S ∗
i=1
Ai
Ai∩Aj=∅
= m ∗ (Q ∩ Ai)+m ∗ (Q ∩ Aj)
� ∞
i=1 Ai ∈ S ∗
= m ∗ (Q ∩ An+1)+m ∗
= m ∗ (Q ∩ An+1)+
=
Pn i=1 Ai∈S∗
= m ∗
�
Q ∩
4.)
=
≥
n →∞
m ∗ (Q) ≥
∞�
i=1
iii)
≥ m ∗
n�
i=1
n�
i=1
n�
i=1
n+1 �
m ∗ (Q ∩ Ai)
Ai
i=1
m ∗ (Q ∩ Ai)+m ∗
m ∗ (Q ∩ Ai)+m ∗
m ∗ (Q ∩ Ai)+m ∗
�
∞�
Q ∩
i=1
Ai
�
Q ∩
n�
i=1
Ai
n�
m ∗ (Q ∩ Ai)
i=1
�
+ m ∗
⎛ �
n�
⎝Q ∩
⎛
⎜ �
⎜ n�
⎜
⎜Q
∩
⎜
⎝ i=1
i=1
Ai
Ai
� C
� �� �
⊃( P ∞
i=1 Ai)C
⎛ �
∞�
⎝Q ∩
⎛ �
∞�
⎝Q ∩
i=1
i=1
Ai
�
+ m ∗
⎛ �
∞�
⎝Q ∩
i=1
Ai
� ⎞
C
⎠
Ai
� ⎞
C
⎠
⎞
⎟
⎠
� ⎞
C
⎠
� ⎞
C
⎠
�

m ∗ (Q)
� ∞
i=1 Ai
�
∞�
Q ⊂ Q ∩
i=1
Ai
� ⎛ �
∞�
+ ⎝Q ∩
i=1
ii)
≤ m ∗
⎛
∞�
�
∞�
⎝Q ∩ Ai + Q ∩
iii)
≤ m ∗
�
Q ∩
m ∗ (Q) =m ∗
i=1
∞�
i=1
Ai
�
∞�
Q ∩
i=1
Ai
Ai
� ⎞
C
⎠ + ∅ + ...
� C
�
+ m ∗
⎛ �
∞�
⎝Q ∩
i=1
Ai
i=1
⎞
+ ∅ + ... ⎠
Ai
�
+ m ∗
⎛ �
∞�
⎝Q ∩
A, B ∈ S ∗ A∩B ∈ S ∗ S ∗
S ∗
Q ⊂ W A ∈ R
S(R) ⊂ S ∗
Q ⊂ Q ∩ A + Q ∩ A C + ∅ + ...
� ⎞
C
⎠ + m ∗ (∅)+...
i=1
m ∗ (Q) = m ∗ (Q ∩ A + Q ∩ A C + ∅ + ...)
≤ m ∗ (Q ∩ A)+m ∗ (Q ∩ A C )+m ∗ (∅)
Q ∩ A ⊂
Q ∩ A C ⊂
∞�
(Ai ∩ A)
i=1
∞�
(Ai ∩ A C )
i=1
Ai
� �� �
=0
Ai ∈ R
� ⎞
C
⎠
+ ...

� ∞
i=1 Ai
m ∗
inf
≤
=
m ∗ (Q ∩ A)+m ∗ (Q ∩ A C )
∞�
∞�
m(Ai ∩ A)+ m(Ai ∩ A C )
i=1
∞�
m(Ai)
i=1
i=1
m ∗ (Q ∩ A)+m ∗ (Q ∩ A C )
�
∞�
∞�
≤ inf m(Ai) : Ai ∈ R, Q ⊂
= m ∗ (Q)
i=1
i=1
m ∗ (Q) =m ∗ (Q ∩ A)+m ∗ (Q ∩ A C )
m ∗ |S ∗ : S∗ → [0, ∞)
∀A ∈ R : A ∈ S ∗
R ⊂ S ∗
S(R) ⊂ S ∗
m ∗ (A) ≥ 0
m ∗ (∅)
m
= 0
∗
�
∞�
�
iii)
≤
∞�
m ∗ (Ai)
Ai
i=1
i=1
Ai
�
S ∗ m ∗

m ∗
≥
� ∞�
i=1
Ai
�
≥
=
=
� ∞
i=1 Ai
∞�
m ∗
⎛
⎜
⎝ Ai
⎞
∞�
⎟
∩ Ai
⎟
i=1 ⎠
� �� �
i=1
∞�
i=1
=Ai
m ∗ (Ai)+m ∗ (∅)
� �� �
=0
∞�
m ∗ (Ai)
i=1
+ m∗
⎛
⎞
⎜ ∞�
�
∞�
�C ⎟
⎜ Ai ∩ Ai
⎟
⎝i=1
i=1 ⎠
� �� �
=∅
m ∗ : S(R) → [0, ∞] m : R → [0, ∞]
m ∗ S(R) ⊂ S ∗
m ∗
A ∈ R m(A) =m ∗ (A)
≤ (A, ∅, ∅,...)
m ∗
m ∗ (A) ≤ m(A)+m(∅ + ...
= m(A)
� ∞
i=1 Ai
⎛
⎞
m(A) ≤
∞�
⎜
⎟
m ⎝ (Ai ∩ A) ⎠
� �� �
i=1
∈R
∞�
≤ m(Ai ∩ A)
≤
i=1
∞�
m(Ai)
i=1
m(A) ≤ m ∗ (A)
�∞ i=1 (A ∩ Ai)

(An)n
An ↑ W
∀n ∈ N : m(An) < ∞
C ∈ S
m(C) < ∞
m(C ∩·):S → [0, ∞),A↦→ m(C ∩ A)
∀A ∈ S : m(C ∩ A) < ∞
m(C ∩∅) = m(∅) =0
�
∞�
�
∞�
m C ∩ = m(C ∩ Ai)
i=1
Ai
m1 m2 S(E)
W ∈ DCn
m1 = m2
(Cn)n
i=1
m(C ∩ A) ≥ 0
m(C ∩ A) ≤ m(C) < ∞
E ⊂ Pot(W )
Cn ↑ W
∀n ∈ N : m1(Cn) =m2(Cn) < ∞
S(E)
⇐⇒
DCn := {B ∈ S(E) :m1(B ∩ Cn) =m2(B ∩ Cn)}
m1(W ∩ Cn) =m1(Cn) Cn∈E
= m2(Cn) =m2(W ∩ Cn)

S(F p )
B ∈ DCn
m(Cn ∩·)
C ∈ E
Bi ∈ DCn
C ∈ E
B C ∈ DCn
m1(B C ∩ Cn) = m1(Cn) − m1(B ∩ Cn)
m1
m1 = m2
� ∞�
i=1
� ∞
Bi ∩ Cn
m1(C ∩ Cn
� �� �
∈E
E
A ∈ S(E)
B∈DCn
= m2(Cn) − m2(B ∩ Cn)
= m2(B C ∩ Cn)
i=1 Bi ∈ DCn
�
m1 =
Bi∈DCn
=
∞�
m1(Bi ∩ Cn)
i=1
∞�
m2(Bi ∩ Cn)
i=1
m2 = m2
C ∩ Cn ∈ E
) m1|E=m2|E
� ∞�
i=1
Bi ∩ Cn
= m2(C ∩ Cn)
� �� �
∈E
E ⊂ DCn
D(E) ⇒ D(E) ⊂ DCn
⇒ S(E) ⊂ DCn
m1(A) = lim
n→∞ m1(A ∩ Cn) S(E)⊂DCn
= lim
n→∞ m2(A ∩ Cn) =m2(A)
(−n, n] ↑ R
l p ((−n, n]) = (2n) p < ∞
�
l p : F p → [0, ∞)

X −1
(W, S), (V,T)
X : W → V
X −1 � B C� =(X −1 (B)) C
X −1 : Pot(V ) → Pot(W )
X−1 ( �
i∈I Bi) = �
i∈I X−1 (Bi)
X−1 ( �
i∈I Bi) = �
i∈I X−1 (Bi)
X−1 ( �
i∈I Bi) = �
i∈I X−1 (Bi)
B1 ⊂ B2
W = X −1 (V )
w ∈ X −1
X −1 (B1) ⊂ X −1 (B2)
w ∈ X −1 (B C ) ⇐⇒ X(w) ∈ B C
B+B C =V
⇐⇒ X(w) �∈ B
� �
i∈I
Bi
�
∀ ∃
∃! ∃
⇐⇒ w �∈ X −1 (B)
X −1 (B)+X −1 (B) C =W
⇐⇒ w ∈ (X −1 (B)) C
⇐⇒ X(w) ∈ �
Bi ⇐⇒ ∃i : X(w) ∈ Bi
i∈I
⇐⇒ ∃i : w ∈ X −1 (Bi) ⇐⇒ w ∈ �
X −1 (Bi)
i∈I

w ∈ X −1 (B1) ⇒ X(w) ∈ B1
B1⊂B2
⇒ X(w) ∈ B2
⇒ w ∈ X −1 (B2)
w ∈ W
∀w ∈ W ∃v ∈ V : X(w) =V
⇒ ∀w∈W ∃v ∈ V : w ∈ X −1 (v)
⇒ ∀w∈W : w ∈ X −1 (V )
⇒ W ⊂ X −1 (V )
⇒ W = X −1 (V )
X : W → V ⇐⇒
∀A ∈ T : X −1 (A) ∈ S
X −1 (T ) ⊂ S X :(W, S) → (V,T)
S ∗ := {B ⊂ V |X −1 (B) ∈ S}
T = S(E)
X : W → V ⇐⇒ ∀A ∈ E : X −1 (A) ∈ S
V ∈ S ∗ X −1 (V )=W ∈ S
A ∈ S ∗ A C ∈ S ∗
Ai ∈ S ∗
X −1 (A C ) i) =( X −1 (A)
� �� �
∈S
� ∞
i=1 Ai ∈ S ∗
X −1
� ∞�
i=1
Ai
�
ii)
=
∞�
i=1
∈S
) C ∈ S
X −1 (Ai)
� �� �
∈ S

⇒ A ∈ E
A ∈ E ⊂ S(E) =T
⇐
∀C ∈ S(E) : X −1 (C) ∈ S
X −1 (A) ∈ S
Def
⇒
∗
E ⊂ S
S(E)
⇒
∗
S(E) ⊂ S
∀A ∈ E : X −1 (A) ∈ S
Def
⇒ ∀A ∈ S(E) : X −1 (A) ∈ S
f : R p → R k (B p , B k )
∀A ∈ B k : X −1 (A) ∈ B p
S( R k )=B k
∀ U : f −1 (U)
∞
±∞
B k
0 ·∞ = 0 = ∞·0
∞ + ∞ = ∞
−∞ − ∞ = −∞
∞·(−∞) = −∞ =(−∞) ·∞
∞·∞ = ∞
(−∞) · (−∞) = ∞
∞ 1 1 1
, , , , ∞−∞
∞ ∞ 0 −∞

R := R ∪{−∞, ∞}
U := {∅, {∞}, {−∞}, {−∞, ∞}}
A + D ∈ B (A + D) C ∈ B
Ai + Di ∈ B
B := {A + D : A ∈ B,D ∈ U}
R = R + {−∞, ∞} ∈ B
(R + {∞, −∞}) \(A + D) =R\A
∞�
∞�
(Ai + Di) =
i=1
i=1
����
∈B
Ai
� �� �
∈B
B E = {[t, ∞] :t ∈ R}
E ⊂ B
∞�
+
+ {∞, −∞}\D ∈ B
� �� �
∈U
i=1
Di
� �� �
∈U
[t, ∞] ={x ∈ R : x ≥ t} ∪ {∞}
S(E) ⊂ B
∈ B
[a, b) = {x ∈ R : a ≤ x

U ⊂ S(E)
B = B + U ⊂ S(E)
f : W → R (S, B) ⇐⇒ ∀t ∈ R : {f ≥ t} ∈S
B E = {[t, ∞] :t ∈ R}
f : W → R
{f ≥ t} = f −1 {[t, ∞]}
{f ≥ t} ∈S ⇐⇒ {f>t}∈S ⇐⇒ {f ≤ t} ∈S ⇐⇒ {fg}, {f t} = f ≥ t +
n=1
1
�
∈ S
n
⇒ {f≤t} = {f >t} C ∈ S
∞�
�
⇒ {f

{f

−g
g + c
f + g ⇐⇒ ∞ − ∞
f −g + t
f · g
f 2
∀t ∈ R : {g ≥ t} ∈S
⇒ ∀t∈R : {g + c ≥ t} = {g ≤ t − c} ∈S
{f = ∞} ∩ {g = −∞} + {f = −∞} ∩ {g = ∞} = ∅
f(x)+g(x) ≥ t ⇐⇒ f(x) ≥−g(x)+t
f − g = f +(−g)
=
{f + g ≥ t} = {f ≥−g + t} ∈S
{f 2 ≤ t} = {f ≤ √ t}∩{f ≥− √ t}∈S
f · g = (f + g)2 +(f − g) 2
4
{f · g ≥ t}∩({f ∈ R}∪{g ∈ R})
� �
2 2
(f + g) +(f − g)
≥ t ∩ ({f ∈ R}∪{g ∈ R}) ∈ S
4
{f · g ≥ t}∩{f = ±∞,g = ±∞}
= ({f = ∞} ∩ {g = ∞}) ∪ ({f = −∞} ∩ {g = −∞}) ∈ S
{f · g ≥ t} = {f · g ≥ t}∩({f ∈ R}∪{g ∈ R})
∪ ({f · g ≥ t}∩{f = ±∞,g = ±∞}) ∈ S
1/g ⇐⇒ {g =0} + {g = ±∞} = ∅

�
∀t ∈ R : {g ≥ t} ∈S ⇒ ∀t∈R : g ≤ 1
�
∈ S
t
� �
1
⇒ ∀t∈R : ≥ t = {g ≤
g 1
t }∈S
� �
sup fn ≤ t
n
� �
inf fn ≥ t
n
� �
sup fn ≤ t
n
� �
inf fn ≥ t
n
f,fn : W → R
=
=
∞�
{fn ≤ t}
n=1
∞�
{fn ≥ t}
n=1
= {w ∈ W : supfn(w)
≤ t}
n
= {w ∈ W : ∀n ∈ N : fn(w) ≤ t}
∞�
= {fn ≤ t}
n=1
= {w ∈ W : inffn(w)
≥ t}
n
= {w ∈ W : ∀n ∈ N : fn(w) ≥ t}
�
∞�
�
= {fn ≥ t}
n=1
sup fn, inf fn, lim sup fn, lim inf
n→∞ n→∞ fn, |f|
lim
n→∞ fn

inf fn
limn→∞ fn
sup fn
R
� �
sup fn ≤ t
n
� �
inf fn ≥ t
n
lim sup fn
n→∞
lim inf
n→∞ fn
Def
∞ −∞
=
=
∞�
{fn ≤ t} ∈S
n=1
∞�
{fn ≥ t} ∈S
n=1
= inf
n sup
m≥n
Def
= sup
n
inf
fm
m≥n fm
|f| = f+ − f−
= max(f,0) − max(−f,0)
lim inf
n→∞ fn =limsupfn
n→∞
lim
n→∞ fn = lim inf
n→∞ fn =limsupfn
n→∞

T :=
� n�
i=1
f :(W, S) → (R, B)
A ∈ S
�
1 w ∈ A
1A : W → R,w ↦→
0
1 −1
A (B) =
B ∈ B
⎧
⎪⎨
⎪⎩
Ai ∈ S
A ∈ S 1 ∈ B,0 �∈ B
A C ∈ S 0 ∈ B,1 �∈ B
W ∈ S 0, 1 ∈ B
∅∈S 0, 1 �∈ B
f :(W, S) → (R, B),w ↦→
(W, S)
� n
i=1 Ai = W 0 ≤ ai < ∞
n�
ai1Ai(w)
i=1
ai1Ai (w) :0≤ ai < ∞,Ai ∈ S, n ∈ N,
1Ai
:(W, S) → (R, B)
ai :(W, S) → (R, B),w ↦→ ai
u, v ∈ T c ∈ [0, ∞)
cu, u + v, max(u, v), min(u, v) ∈ T
n�
�
Ai = W
i=1

⇒
un
n�
i=1 j=1
f =supfn
un
u = � n
i=1 ai1Ai
cu =
n�
i=1
k�
�
n�
(Ai ∩ Bj) =
u + v =
max(u, v) =
min(u, v) =
i=1
n�
v = � k
j=1 bj1Bj
Ai
i=1 j=1
n�
i=1 j=1
n�
i=1 j=1
cai1Ai (w) ∈ T
� ⎛
k�
∩ ⎝
Bj
j=1
⎞
⎠ = W ∩ W = W
k�
(ai + bj)1Ai∩Bj ∈ T
k�
max(ai,bj)1Ai∩Bj ∈ T
k�
min(ai,bj)1Ai∩Bj ∈ T
T ∗ = {f : W → [0, ∞] }
±∞
f ∈ T ∗ ⇐⇒ ∃ (un)n T : f =supun
n
⇐ un ≥ 0 sup n un sup n un ≥ 0
Ai,n =
un =
n2 n �−1
i=0
n
�
i +1
≤ f

un ∈ T
un
�
i +1
≤ f

f(w) =∞
∃n0 ∀n ≥ n0 : |un(w) − f(w)| ≤ 1
2 n
lim
n→∞ un(w) = f(w)
∀n ∈ N : w ∈{f ≥ n}
lim
n→∞ un(w) = lim n = ∞ = f(w)
n→∞
∀w ∈ W : lim
n→∞ un(w) =f(w)
f,g ∈ T ∗ c ∈ [0, ∞)
cf, f + g, f · g, min(f,g), max(f,g) ∈ T ∗

m : S → [0, ∞]
1A � n
i=1 ai1Ai
u =
� n �
n�
i=1
i=1
�
u =
Ai =
S ⊂ Pot(W )
1A
1Adm := m(A)
ai1Ai dm :=
n�
i=1
n�
i=1 j=1
ai1Ai =
u = � n
i=1 ai1Ai ∈ T
n�
aim(Ai)
i=1
k�
j=1
bj1Bj
k�
(Ai ∩ Bj) =
k�
j=1
w ∈ Ai ∩ Bj ⇒ ai = u(w) =bj
n�
k�
i=1 j=1
ai1Ai∩Bj =
n�
k�
i=1 j=1
Bj
bj1Ai∩Bj

n�
aim(Ai)
i=1
P k
j=1 Bj=W
=
=
=
=
P n
i=1 Ai=W
=
⎛
n�
aim ⎝Ai ∩
i=1
n�
i=1 j=1
k�
j=1 i=1
k�
j=1
Bj
k�
aim(Ai ∩ Bj)
n�
bjm(Ai ∩ Bj)
k�
�
n�
bjm
j=1
i=1
k�
bjm(Bj)
j=1
Ai ∩ Bj
u, v ∈ T c ∈ R
�
�
cudm = c udm
�
� �
(u + v)dm = udm + vdm
� �
u ≤ v
udm ≤ vdm
Ai ∩ Aj = ∅
�
udm =
n�
i=1
u = � n
i=1 ai1Ai
u =
v =
ai
�
1Aidm =
⎞
⎠
�
u = � n
i=1 ai1Ai ∈ T
n�
aim(Ai)
i=1
v = � k
j=1 bj1Bj
n�
k�
i=1 j=1
n�
k�
i=1 j=1
ai1Ai∩Bj
bj1Ai∩Bj

u ≤ v
=
=
=
=
=
=
=
�
cudm =
�
(u + v)dm
� ⎛
n� k�
⎝
� n �
n�
�
�n
c · ai1Ai dm
i=1
n�
�
= c aim(Ai) =c udm
i=1
n�
k�
ai1Ai∩Bj + bj1Ai∩Bj
i=1 j=1
i=1 j=1
i=1 j=1
i=1 j=1
n�
i=1 j=1
k�
(ai + bj)1Ai∩Bj dm
k�
(ai + bj)m(Ai ∩ Bj)
k�
aim(Ai ∩ Bj)+
n�
aim(Ai ∩
i=1
n�
aim(Ai)+
i=1
k�
j=1
Bj
� �� �
=W
)+
n�
i=1 j=1
⎞
⎠ dm
k�
bjm(Ai ∩ Bj)
k�
bjm(Bj ∩
j=1
n�
bjm(Bj)
j=1
� �
udm + vdm
w ∈ Ai ∩ Bj ⇒ ai ≤ bj
n�
i=1
Ai
� �� �
=W
)

�
udm =
u ≤ limn→∞ un
un,u
≤
=
n�
i=1 j=1
n�
i=1 j=1
�
k�
aim(Ai ∩ Bj)
k�
bjm(Ai ∩ Bj) =
vdm
u, un ∈ T un
�
�
udm ≤ lim
n→∞
undm
u = � k
i=1 ai1Ai ∈ T c ∈ (0, 1)
un ≥ 0
cu < limn→∞ un
un ≥ cu1Bn =
Bn := {un ≥ c · u} ∈S
k�
bjm(Bj)
j=1
� cu un(w) ≥ cu(w)
0 un(w)

Bn ↑ W
Ai ∩ Bn ⊂ Ai ∩ Bn+1
∞�
∞�
(Ai ∩ Bn) =Ai ∩ Bn = Ai ∩ W = Ai
n=1
n=1
Ai ∩ Bn ↑ Ai
m(Ai) Ai∩Bn↑Ai
= lim
n→∞ m(Ai ∩ Bn)
�
k�
udm = aim(Ai) = lim
�
i=1
= lim
n→∞
�
= lim
n→∞
� k �
undm un≥cu1Bn
≥
�
lim
n→∞
c ↑ 1
(un)n
i=1
n→∞
i=1
k�
aim(Ai ∩ Bn)
ai1Ai∩Bn dm = lim
n→∞
u1Bn dm
�
�
undm ≥ c lim
n→∞
�
lim
n→∞
limn→∞ un =limn→∞ vn
n ∈ N
lim
n→∞
�
(vn)n
�
c · u1Bndm = c
�
u1Bndm = c
�
undm ≥ udm
�
undm = lim
n→∞
un ≤ lim
n→∞ vn
vn ≤ lim
n→∞ un
vndm
� k �
i=1
u1Bndm
udm
ai1Ai
1Bndm

�
�
∀n ∈ N :
�
undm ≤ lim
n→∞
�
vndm
∀n ∈ N : vndm ≤ lim
n→∞
undm
�
lim
n→∞
�
undm ≤ lim
n→∞
�
vndm ≤ lim
n→∞
undm
f ∈ T ∗ (un)n un ↑ f
�
�
fdm = lim
n→∞
undm
f,g ∈ T ∗ c ∈ [0, ∞)
�
�
cfdm = c fdm
�
� �
(f + g)dm = fdm+ gdm
� �
f ≤ g
fdm≤ gdm
un ↑ f
un ∈ T un ↑ f 0 ≤ un ≤ f
c ≥ 0
cun ↑ cf
�
cfdm cun↑cf
�
= lim
n→∞
0 ≤ cun ≤ cf
lim
n→∞ cun = c lim
n→∞ un = cf
�
cundm = c lim
n→∞
f ∈ T ∗ (un)n
undm un↑f
�
= c
fdm

un,vn ∈ T un ↑ f,vn ↑ g un + vn ∈ T
�
0 ≤ un + vn ≤ un+1 + vn+1 ≤ f + g
lim
n→∞ (un + vn) = lim
n→∞ un + lim
n→∞ vn = f + g
un + vn ↑ f + g
(f + g)dm un+vn↑f+g
= lim
∀n ∈ N :
�
�
(un + vn)dm
n→∞
�� � �
= lim
n→∞
�
undm + vndm
�
= lim
n→∞
�
undm + lim
n→∞
�
vndm
fdm+ gdm
un↑f,vn↑g
=
�
un ≤ f ≤ g = lim
n→∞ vn
�
undm ≤ lim
n→∞
fdm Def
�
= lim
n→∞
vndm Def
=
�
�
undm ≤ gdm
gdm

ui,n
ui,n
fn ∈ T ∗
(fn)n
f := limn→∞ fn ∈ T ∗
�
lim
n→∞ fndm
�
= lim
n→∞
fndm
fn ui,n ∈ T ui,n ↑ fn
vn := max(un,1,...,un,n) ∈ T
i
un+1,i ≥ un,i
vn = max(un,1,...,un,n)
T ∗
≤ max(un+1,1,...,un+1,n,un+1,n+1)
= vn+1
fn n k ≤ n
un,k ≤ fk ≤ fn ≤ f
vn = max(un,1,...,un,n) ≤ fn ≤ f
∀i ≥ n : ui,n ≤ max(ui,1,...,ui,i) =vi
fn
vn ↑ f
�
fdm vn↑f
�
= lim
n→∞
Vor
= lim
i→∞ ui,n ≤ lim
i→∞ vi
lim
n→∞ fn ≤ lim
n→∞ vn
vn≤fn
≤ lim
n→∞ fn
vndm vn≤fn
�
≤ lim
n→∞
fndm fn≤f
≤
�
fdm

(fn)n T ∗ � ∞
n=1 fn ∈ T ∗
�
�∞
∞�
�
fndm = fndm
n=1
n=1
�k limk→∞ n=1 ≥ 0 T ∗
� ∞ �
n=1
fndm Def
=
� k�
n=1
�
= lim
f ≥ 0
fn
�
k
↑
k�
∞�
n=1
fn
lim fndm
k→∞
n=1
↑ = lim
k→∞
k�
�
k→∞
n=1
f ∈ T ∗
�
Q : S → [0, ∞],A↦→
A
fndm Def
=
n=1
�
�k
fndm
n=1
∞�
�
fndm
�
fdm = 1Afdm

Q(A) =
Q(∅) =
�
∞�
�
Q
=
i=1
Ai
=
�
1Af dm ≥ 0
����
�
≥0
�
1∅fdm = 0dm =0
�
1 P ∞
i=1 Aifdm
� � ∞ �
i=1
1Ai
�
fdm
=
�
n�
lim 1Aif dm
n→∞
i=1
� �� �
,≥0
=
�
lim
n→∞
�n
1Aifdm = lim
n→∞
=
i=1
∞�
Q(Ai)
i=1
n�
�
1Aifdm
i=1

⇐⇒
|f|
⇒
�
�
f :(W, S) → (R, B)
f + dm < ∞
f − dm < ∞
f = f + − f −
� �
fdm := f + �
dm − f − dm
f : W → R
|f| |f| = f + + f −
�
�
|f|dm =
(f + + f − )dm =
⇒ g = |f| |f| ≤g
⇒
f + ,f − ≤|f| ≤g
�
f + � �
dm ≤ |f|dm ≤
�
f − � �
dm ≤ |f|dm ≤
�
|f| ≤g
f + �
dm +
gdm Vor
< ∞
gdm Vor
< ∞
f,g
cf, f + g, max(f,g), min(f,g)
c ∈ R
�
�
cfdm = c fdm
�
� �
(f + g)dm = fdm+ gdm
f − dm < ∞

�
|f|dm < ∞
� �
|cf|dm =
�
|c||f|dm = |c|
|f|dm < ∞
| max(f,g)| ≤ max(|f|, |g|) ≤|f| + |g|
| min(f,g)| ≤ min(|f|, |g|) ≤|f| + |g|
|f + g| ≤ |f| + |g|
�
�
(|f| + |g|)dm =
�
|f|dm +
(cf) + = cf + (cf) − = cf −
|g|dm < ∞
�
�
(cf)dm = cf + �
dm − cf − dm
�
= c f + �
dm − c f − �
dm = c fdm
�
cf = c(f + − f − )=(−c)f − − (−c)f +
cfdm Def
=
�
�
= −c
= c
= c
�
cfdm =
(−c)f − �
dm − (−c)f + dm
�
f − �
dm − (−c) f + dm
��
f + �
dm − f − �
dm
�
fdm
�
0dm =0=0·
f + g = f + + g + − (f − + g − )
fdm

�
(f + g)dm =
f ≤ g
�
−f |f|
f,g
f ≤ g
=
=
�
�
�
fdm =
�
�
−
(f + + g + �
)dm −
f + �
dm + g + dm −
�
fdm+ gdm
� �
fdm≤ gdm
��
�
� �
�
� fdm�
� ≤
�
|f|dm
f + ≤g +
≤
f − ≥g −
≤
fdm ≤
fdm =
f + ≤ g +
f − ≥ g −
(f − + g − )dm
�
f − �
dm − g − dm
�
f + �
dm − f − dm
�
g + �
dm − f − dm
�
g + �
dm − g − �
dm = gdm
f ≤ |f|
−f ≤ |f|
�
�
|f|dm
�
−fdm≤ |f|dm

��
�
� �
�
� fdm�
� ≤
�
|f|dm

(fn)n
g, fn
lim infn→∞ fn
(fn)n
lim supn→∞ fn
�
(infk≥n fk)n
n ∈ N
∀n ∈ N : fn ≥ 0
�
lim inf
n→∞ fndm
�
≤ lim inf fndm
n→∞
fn ≥ g
�
lim inf
n→∞ fndm
�
≤ lim inf fndm
n→∞
∀n ∈ N : fn ≤ 0
�
�
lim sup fndm ≥ lim sup
n→∞
n→∞
fndm
∀k ≥ n :
lim sup
n→∞
fn ≤ g
�
fndm ≥ lim sup
n→∞
∀k ≥ n :inf
k≥n fk ≤ fk
�
�
fndm
inf
k≥n fkdm
�
≤ fkdm
inf
k≥n fkdm
�
≤ inf
k≥n
fkdm
inf{fk(x) :k ≥ n} ≤inf{fk(x) :k ≥ n +1}
�
lim
n→∞ inf
k≥n fkdm
�
= lim
n→∞
inf
k≥n fkdm
≤ lim
n→∞ inf
�
k≥n
�
fkdm
= liminf
k→∞
fkdm

fn − g ≥ 0
� gdm
�
=
=
a)
�
lim inf
n→∞ fndm
�
− gdm
� �� �
� �
∈R
lim inf
n→∞ fn
�
�
− g dm
lim inf
n→∞ (fn − g)dm
� �� �
≤ lim inf
n→∞
= liminf
n→∞
≥0
�
(fn − g)dm
� �
fndm − gdm
� �� �
∈R
lim inf
n→∞ fndm
�
≤ lim inf
n→∞
− lim sup fn = − lim
n→∞
= lim
n→∞
fndm
n→∞ sup fm
m≥n
�
− sup fm
m≥n
= lim
n→∞ inf
m≥n (−fm)
= liminf
n→∞ (−fn)
−fn ≥ 0
�
�
− lim sup fn dm
n→∞
� �� �
= lim inf
n→∞
≤0
(−fn)dm
−fn≥0
≤
�
lim inf (−fn)dm
n→∞
�
= − lim sup
n→∞
fndm
�

lim infn→∞(−fn) =− lim supn→∞ fn
−fn ≥−g
−g
�
�
− lim sup fndm
n→∞
= lim inf
n→∞ (−fn)dm
−fn≥−g
≤
�
lim inf (−fn)dm
n→∞
�
= − lim sup
n→∞
fndm
g
g �
limn→∞ fndm
fn limn→∞ fn = f |fn| ≤
� �
lim
n→∞
fndm = lim
n→∞ fndm
|fn| ≤g fn lim infn→∞ fn
�
lim
n→∞ fndm
�
= lim inf
n→∞ fndm
fn≥−g
≤
�
lim inf fndm
n→∞
�
≤
fn≤g
≤
lim sup fndm
n→∞
�
lim sup fndm
n→∞
�
= lim
n→∞ fndm

∃N ∈ S
�
�
1.) m(N) =0
2.) {w ∈ W : }⊂N
⇐⇒
f,g : (W, S) → (R, B) f = g
� �
fdm = gdm
N := {f �= g} ∈S
m(N) = 0
1 N C f = 1 N C g
0 ≤ m(A ∩ N) ≤ m(N) =0
m(Ai) = m(Ai ∩ N) +m(Ai ∩ N
� �� �
=0
C )
= m(Ai ∩ N C )
f,g ≥ 0 (un)n, (vn)n un ↑ f vn ↑ g
un1 N C dm =
=
=
vn1 N C dm =
=
=
� kn�
i=1
i=1
ai,n1Ai,n1N C dm =
kn�
i=1
ai,n
kn�
ai,nm(Ai ∩ N C kn�
)= ai,nm(Ai)
�
undm
� kn�
i=1
i=1
i=1
bi,n1Bi,n1N C dm =
kn�
i=1
bi,n
kn�
bi,nm(Bi,n ∩ N C kn�
)= bim(Bi)
�
vndm
i=1
�
�
1 Ai,n∩N C dm
1 Bi,n∩N C dm

⇒
un1 N C ↑ f1 N C vn1 N C ↑ g1 N C
�
�
fdm =
=
lim
n→∞
�
�
= lim
n→∞
�
= gdm
undm s.o.
= lim
n→∞
f1 N C dm Vor
=
�
�
g1 N C dm
vn1 N C dm Vor
= lim
n→∞
f,g f + ,g + f − ,g −
f :(W, S) → (R, B)
�
|f|dm =0 ⇐⇒ f =0m −
⇐ g =0 |f| = g
� � �
|f|dm = gdm = 0dm =0
��
m |f| ≥ 1
��
n
=
n|f|≥1
≤
=
�
�
1 {|f|≥ 1
n }dm
n|f|1 {|f|≥ 1
n }dm +
un1 N C dm
�
�
vndm
�
�
n|f|dm = n |f|dm =0
�
∞� �
m({|f| > 0}) = m f ≥
n=1
1
�
n
�
∞�
��
≤ m f ≥ 1
��
=0
n
n=1
N := {f : W → R : f =0 }
n|f|1 {|f|< 1
n }dm

0:W → R
c =0 cf =0
c �= 0
m(cf �= 0)=m(∅) =0
m(cf �= 0) c�=0
= m(f �= 0)=0
f,g ∈ N
f + g �= 0 f �= 0 g �= 0
{f + g �= 0}⊂{f �= 0}∪{g �= 0}
m(f + g �= 0)≤ m(f �= 0)+m(g �= 0)=0

˜L 1 :=
L 1
�
�
f :(W, S) → (R, B) :
�
|f|dm < ∞
�·�1: ˜ L 1 �
→ [0, ∞),f ↦→ |f|dm
� f + g �1 ≤ � f �1 + � g �1
� cf �1 = |c| �f �1
0 ≤ � |f|dm < ∞
|f + g| ≤|f| + |g|
�
�·�1
� f + g �1 = |f + g|dm
�
≤ (|f| + |g|)dm
� �
= |f|dm + |g|dm
= � f �1 + � g �1
�
�
� cf �1 = |cf|dm = |c| |f|dm = |c| �f �1
N :=
�
f ∈ ˜ L 1 �
:
˜L 1
�
|f|dm =0
= {f =0 }
L 1 := {{f + n : n ∈ N} : f ∈ ˜ L 1 }

{f + n : n ∈ N} = {g + n : n ∈ N} ⇐⇒ f = g
f = {f + n : n ∈ N}
�·�1: L 1 �
→ [0, ∞),f ↦→ |f|dm
f =0 ⇐⇒
⇐⇒ ∃n ∈ N : f = g + n
�
|f|dm =0
{f =0 } = {f ∈ ˜ L 1 �
: |f|dm =0}
˜L 1
f = f
f = g ⇒ g = f
f = g, g = h ⇒ f = h
f = f n =0∈ N
f = g ⇒ g = f f = g + n −n ∈ N g = f +(−n)
f = g, g = h ⇒ f = h
f = g + n1
g = h + n2
f = h + n1 + n2
� �� �
∈N
f + n1 + g + n2 = f + g
� �� �
∈ ˜ L 1
+ n1 + n2
� �� �
∈N
c(f + n1) = cf + cn1
���� ����
∈N
∈ ˜ L 1
L 1

L 1
�·�1
{f + n : n ∈ N} + {g + n : n ∈ N} := {f + g + n : n ∈ N}
c ·{f + n : n ∈ N} := {cf + n : n ∈ N}
f ∈ L 1 n ∈ N
�·�1
∀f ∈ L 1 �
:0≤ |f|dm < ∞
� f �1 = � f + n − n �1
≤ � f + n �1 + � n �1
� �� �
=0
≤ � f �1 + � n �1 =� f �1
� �� �
=0
∀f ∈ ˜ L 1 ∀n ∈ N : � f + n �1=� f �1
�·�1
f,n1,g,n2 ∈ ˜ L 1 f + n1,g+ n2 ∈ ˜ L 1
L 1
� c(f + n1) �1 = |c| �f + n1 �1
� f + n1 + g + n2 �1 ≤ � f + n1 �1 + � g + n2 �1
� f + n1 �1= 0 ⇐⇒ � f �1= 0 ⇐⇒
⇐⇒ f =0
�
|f|dm =0
fn,f ∈ L 1 (fn)n L 1 ⇐⇒
lim
n→∞ � f − fn �1= 0
(fn)n L1 (fnk )k
f ∈ L1 → f
fnk

�
� ∞�
�
�
�
k=1
f :=
(fn)n
|fnk+1
fn1
∀k ≥ 1 ∃nk ∈ N ∀n, m ≥ nk : � fn − fm �1≤ 1
2 k
nk+1 ≥ nk
�
�
�
− fnk | �
� 1
∞�
k=1
∞�
k=1
(fnk )k
=
=
≤
|fnk+1
|fnk+1
j�
k=1
� limk→∞ fnk = � ∞
0
fn1 ∈ L1
�
�∞
∞�
�
|fnk+1 − fnk | dm =
� �� �
k=1
k=1
≥0
∞�
k=1
∞�
k=1
� fnk+1 − fnk �1
1
=1< ∞
2k − fnk | ∈ L1
− fnk | < ∞
∞�
k=1
(fnk+1
(fnk+1
− fnk ) ∈ L1
− fnk )=fnj+1 − fn1
(fnk+1 k=1 − fnk )+fn1
f ∈ L 1
fnk − fn1
|fnk+1
− fnk |dm

L 1
L 1 L 1
fn :(W, S, m) → (R, B) limn→∞ fn = f
|fn| ≤g g ∈ L 1
f ∈ L 1
lim
n→∞ � f − fn �1 = 0
L 1
|fn| ≤g fn, limn→∞ fn
N := {w ∈ W : fn(w) f(w)}
Nn := {w ∈ W : fn >g}
∞�
M := N ∪
n=1
Nn
m(N) =0=m(Nn)
0 ≤
�
∞�
m(M) =m N ∪
g1 M C
≤ m(N)+
n=1
Nn
∞�
m(Nn) =0
n=1
1M C | lim
n→∞ fn| ≤ g1M C
1M C |fn − lim
n→∞ fn| ≤ 2g1M C
lim
n→∞ fn1M C = f1M C
lim
n→∞ |f − fn|dm
�
= lim
n→∞
�
=
=
lim
�
�
|f − fn|1 M C dm
n→∞ |f − fn|1 M C dm
01 M C dm =
�
0dm =0

limn→∞ fn = f f ∈ L 1
limn→∞ � fn − f �1= 0
W =[0, 1],S = B| [0,1],m= l
fn :([0, 1], B| [0,1]) → (R, B),t↦→ n 2 1 [0, 1
n ](t)
lim
n→∞ fn =0
lim
n→∞ � fn − f �1 =
=
�
lim |fn − f|dm
n→∞
lim
n→∞ n2 ��
l 0, 1
��
= lim n = ∞
n n→∞
(fn)n 0| [0,1] fn
f,fn ∈ T ∗
lim
n→∞
lim
n→∞ fn
�
= f
�
fndm = fdm
limn→∞ � fn − f �1= 0
�
lim |fn − f|dm =0
n→∞
�
2
�
fdm = lim inf
n→∞ (fn + f −|fn − f|)dm
�
≤ lim inf
n→∞
�
(fn + f −|fn − f|)dm
�
Vor
= 2 fdm− lim sup
n→∞
�
|fn − f|dm
0 ≤ −lim sup
n→∞
|fn − f|dm ≤ 0

�
�
�
˜L 2 :=
f,g ∈ L 2
L 2
|cf| 2 dm =
�
�
f : W → R :
|f| 2 �
dm < ∞
�
2 |f| dm < ∞
�
|c| 2 |f| 2 dm = |c| 2
�
|f| 2 dm < ∞
2f 2 +2g 2 − (f + g) 2 = f 2 − 2fg + g 2
|f + g| 2 �
dm ≤ 2
f,g ∈ L2 �
�
|fg|dm ≤
� |g| 2 dm =0
|f + g| 2 ≤ 2f 2 +2g 2
= (f − g) 2 ≥ 0
|f| 2 �
dm +2
|f| 2 �
dm
|g| 2 dm =0 ⇐⇒ |g| 2 =0
�
⇐⇒ |g| =0
�
|fg|dm =0=
⇒ |f||g| =0
|f| 2 �
dm
|g| 2 dm
|g| 2 dm < ∞
|g| 2 dm

�
2 |g| dm > 0
�
h :[0, ∞) → R,t↦→ (|f| + t|g|) 2 dm
0 ≤
=
h(t) =
�
|f| 2 �
dm +2t
h ′ �
(t0) = 2 |fg|dm +2t0
t0 =
�
|fg|dm
− �
|g| 2dm h ′′ �
(t0) = |g| 2 dm > 0
t0
�
�
|f| 2 dm − 2
|f| 2 dm −
|fg|dm + t 2
�
�
|g| 2 dm
|g| 2 dm ! =0
�� �2 �� �2 �
|fg|dm |fg|dm
� +
|g| 2 �� �
dm |g| 2
2
dm
�� |fg|dm � 2
� |g| 2 dm
�
2 |g| dm
�
�
|fg|dm ≤
|g| 2 �
dm
�·�2: ˜ L 2 ��
→ [0, ∞),f ↦→
|f| 2 dm
|f| 2 � 1
2
dm
� f + g �2 ≤ � f �2 + � g �2
� cf �2 = |c| �f �2
|g| 2 dm

0 ≤ � |f| 2dm < ∞
�
�·�2
= |f + g| 2 dm
�
≤ (|f| + |g|) 2 dm
�
≤ |f| 2 � �
dm +2 |fg|dm +
� f + g � 2 2
� cf � 2 �
2=
s.o.
≤ � f � 2 2 +2 � f �2� g �2 + � g � 2 2
= (� f �2 + � g �2) 2
N :=
|cf| 2 dm = |c| 2
�
�
f ∈ ˜ L 2 �
:
|g| 2 dm
|f| 2 dm = |c| 2 � f � 2 2
˜L 2
|f| 2 �
dm =0
= {f =0 }
L 2 := {{f + n : n ∈ N} : f ∈ ˜ L 2 }
{f + n : n ∈ N} = {g + n : n ∈ N} ⇐⇒ ∃n∈N : f = g + n
⇐⇒ f = g
�·�2: L 2 ��
→ [0, ∞),f ↦→
L 2
|f| 2 � 1
2
dm
f =0 ⇐⇒ |f| 2 =0
�
⇐⇒ |f| 2 dm =0
⇐⇒ � f �2= 0

�
{f =0 } = f ∈ ˜ L 2 �
:
L 2
f = f
f = g ⇒ g = f
f = g, g = h ⇒ f = h
|f| 2 �
dm =0
f = f n =0∈ N
f = g ⇒ g = f f = g + n −n ∈ N g = f +(−n)
f = g, g = h ⇒ f = h
L 2
f = g + n1
g = h + n2
f = h + n1 + n2
� �� �
∈N
f + n1 + g + n1 = f + g
� �� �
∈ ˜ L 2
∈ ˜ L 2
+ n1 + n2
� �� �
∈N
c(f + n1) = cf + cn1
���� ����
∈N
{f + n : n ∈ N} + {g + n : n ∈ N} := {f + g + n : n ∈ N}
c ·{f + n : n ∈ N} := {cf + n : n ∈ N}
f ∈ ˜ L 2 n ∈ N
� f �2 = � f + n − n �2
≤ � f + n �2 + � n �2
� �� �
=0
≤ � f �2 + � n �2 =� f �2
� �� �
=0
L 2

�·�2
�·�2
� f + n �2=� f �2
L 2
f + n1,g+ n2 ∈ ˜ L 2
� c(f + n1) �2 = |c| �f + n1 �2
� f + n1 + g + n2 �2 ≤ � f + n1 �2 + � g + n2 �2
� f + n1 �2= 0 ⇐⇒ � f �2= 0 ⇐⇒
�
� ∞� �
�
�
gk
�
k=1 2
0 ≤ gk ∈ L 2
�
�
�
�
⇐⇒ |f| 2 =0
⇐⇒ f =0
�
|f| 2 dm =0
fn,f ∈ L 2 (fn)n L 2 ⇐⇒
lim
n→∞ � f − fn �2= 0
�
� ∞� �
�
�
�
�
�
�
gk
�
k=1 2
=
x→x 2
=
=
≤
∞�
k=1
⎛
�
⎝
� �
���
lim
⎛
⎜�
⎜
⎝
⎛
⎝ lim
n→∞
� gk �2
n�
gk
n→∞
k=1
� ⎞
2 �
�
� dm⎠
�
�
n�
�2 lim gk
n→∞
k=1
� �� �
≥0, ↑
�2 � � n �
k=1
gk
1
2
⎞
⎟
dm ⎟
⎠
⎞
dm⎠
1
2
1
2

(fn)n
�
� ∞�
�
�
�
�
�
�
�
gk
�
k=1 2
x→ √ x
= lim
n→∞
⎛
�
⎝
� � n�
���
�
� n�
�
= lim �
n→∞ �
≤ lim
n→∞
k=1
k=1
�
�
�
�
gk
�
k=1 2
gk
� ⎞
2 �
�
� dm⎠
�
n�
∞�
� gk �2=
k=1
1
2
� gk �2
(fn)n L2 (fnk )k
f ∈ L2 → f
fnk
fn1
∀k ≥ 1 ∃nk ∈ N ∀n, m ≥ nk : � fn − fm �2≤ 1
2 k
nk+1 ≥ nk
�
� ∞� �
�
�
k=1
|fnk+1
∞�
k=1
∞�
k=1
|fnk+1
|fnk+1
j�
k=1
�
�
�
− fnk | �
� 2
≤
≤
∞�
k=1
∞�
k=1
− fnk | ∈ L2
− fnk | < ∞
∞�
k=1
(fnk+1
(fnk+1
− fnk ) ∈ L2
� fnk+1 − fnk �2
1
=1< ∞
2k − fnk )=fnj+1 − fn1
fnk − fn1

f :=
(fnk )k
� limk→∞ fnk = � ∞
0
fn1 ∈ L2
(fnk+1 k=1 − fnk )+fn1
f ∈ L 2
L 2 L 2
f ∈ L 2

(Kn)n
� b
a
� b
a
xn,j
g :[a, b] ⊂ R → R
∃h ∈ L1 [a, b] h = g
� b �
g(x)dx =
a
[a,b]
� b
a g(x)dx
�
hdl := hdl| [a,b]
cnj := inf{g(x) :x ∈ (xn,j−1,xnj )}
dnj := sup{g(x) :x ∈ (xn,j−1,xnj )}
Un :=
On :=
kn�
cn,j(xn,j − xn,j−1) ↗
j=1
kn�
dn,j(xn,j − xn,j−1) ↘
j=1
un :=
on :=
kn�
j=1
kn�
j=1
g(x)dx = lim
n→∞ Un = lim
n→∞
= lim
kn�
cn,j
n→∞
j=1
g(x)dx = lim
n→∞ On = lim
n→∞
= lim
kn�
dn,j
n→∞
j=1
�
�
� b
a
� b
a
cn,j1 (xn,j−1,xn,j)(x)
dn,j1 (xn,j−1,xn,j)(x)
kn�
j=1
g(x)dx
g(x)dx
cn,jl[(xn,j−1,xn,j)]
M>0
[a, b]
�
1 (xn,j−1,xn,j)(x)dl = lim
n→∞
kn�
j=1
dn,jl[(xn,j−1,xn,j)]
�
1 (xn,j−1,xn,j)(x)dl = lim
n→∞
[a,b]
[a,b]
undl
ondl

on ≥ un
� b
a
�
�
g(x)dx =
|un|, |on| ≤ M1 [a,b]
M1 [a,b]dl = M(b − a) < ∞
[a,b]
limn→∞ on≥limn→∞ un
=
=
lim
n→∞ undl
�
=
�
�
�
= 0
[a,b]
[a,b]
[a,b]
lim
n→∞ on = lim
n→∞ un
limn→∞ un ≤ g ≤ limn→∞ un
h := limn→∞ on
lim
n→∞ on = g = lim
n→∞ un
lim
[a,b]
n→∞ ondl
�
�
� lim
n→∞ on − lim
�
lim
n→∞ on − lim
lim
n→∞ ondl
�
−
n→∞ un
n→∞ un
�
�
� dl
�
dl
lim
[a,b]
n→∞ undl
g : R → [0, ∞) [−n, n],n∈ N
� ∞
� n
lim g(x)dx
n→∞
−n
∃h ∈ L 1 (R) h = g
−∞
� n �
g(x)dx := lim
n→∞
g(x)dx =
−n
hdl
R
� ∞
g(x)dx < ∞ ⇐⇒ h l −
−∞

[−n, n] ↑ R
� n
�
lim
n→∞
g(x)dx
−n
= lim
n→∞
=
=
�
R
g 1 [−n,n]
� �� �
≥0,
g lim
n→∞ 1 [−n,n]dl
�
gdl
dl

Ai ∈ Si
Si
(Wi,Si,mi) Wi
mi
W := W1 × ...× Wn
m A1 × ...× An
m(A1 × ...× An) =m1(A1) · ...· mn(An)
A1 × ...× An
S1 ⊗ ...⊗ Sn := S({W1 × ...× Wi−1 × Ai × Wi+1 × ...× Wn : Ai ∈ Si})
⊃
⊂
Ei Si 1 ≤ i ≤ n
(Eik)k Ei Eik ↑ Wi
B1 × ...× Bn =
=
S1 ⊗ ...⊗ Sn = S({B1 × ...× Bn : Bi ∈ Ei})
n�
W1 × ...× Wi−1 × Bi × Wi+1 × ...× Wn
� �� �
i=1
∈ S1 ⊗ ...⊗ Sn
∈S1⊗...⊗Sn
W1 × ...× Wi−1 × Bi × Wi+1 × ...× Wn
∞�
E1k × ...× Ei−1,k × Bi × Ei+1,k × ...× En,k
� �� �
k=1
∈S({B1×...×Bn:Bi∈Ei})
∈ S({B1 × ...× Bn : Bi ∈ Ei})

(Ei,k)k
Ei
Ei,k ↑ Wi
mi(Ei,k) < ∞
Ei
(W1 × ...× Wn,S1 ⊗ ...⊗ Sn)
∀Ai ∈ Ei : m(A1 × ...× An) =m1(A1) · ...· mn(An)
A, B ∈ E
A ∩ B ∈ E
Ei,k ↑ Wi
k ≥ max n i=1 ki
E := {A1 × ...× An : Ai ∈ Ei}
(A1 × ...× An) ∩ (B1 × ...× Bn)
= (A1∩B1) × ...× (An ∩ Bn) ∈ E
� �� � � �� �
∈E1
∈En
w1 × ...× wn ∈ W1 × ...× Wn
∀1 ≤ i ≤ n ∃ki ∀k ≥ ki : wi ∈ Ei,k
w1 × ...× wn ∈ E1,k × ...× En,k
E1k × ...× En,k ↑ W1 × ...× Wn
S(E1 × ...× En)
Q ∈ S1 ⊗ S2 S1 S2
Q ∈ S1 ⊗ S2
Qw1 := {w2 ∈ W2 :(w1,w2) ∈ Q}
Qw2 := {w1 ∈ W1 :(w1,w2) ∈ Q}
� ∞�
k=1
Qk
(Qwi )C =(Q C )wi
�
|wi =
∞�
(Qk)wi
k=1
∀w1 : Qw1 ∈ S2
∀w2 : Qw2 ∈ S1

W ∈ S ′
{(w1,w2) ∈ Q} + {(w1,w2) ∈ W \Q} = W1 × W2
{(w1,w2) ∈ Q}|w1 + {(w1,w2) ∈ Q C }|w1 = (W1 × W2)|w1
(W \Q)w1
� ∞�
w1 ∈ W1
k=1
Qk
�
w1
Q ∈ S ′ Q C ∈ S ′
Qi ∈ S ′
A1 × A2 ∈ S ′
� ∞
Def
= {w2 ∈ W2 :(w1,w2) �∈ Q}
w1 = W2\{w2 ∈ W2 :(w1,w2) ∈ Q}
= W2\Qw1
=
�
w2 ∈ W2 :(w1,w2) ∈
∞�
k=1
Qk
= {w2 ∈ W2 : ∃k :(w1,w2) ∈ Qk}
∞�
= {w2 ∈ W2 :(w1,w2) ∈ Qk}
=
k=1
∞�
k=1
(Qk)w1
S ′ := {Q ⊂ W : Qw1
∈ S2}
(W1 × W2)w1 = W2 ∈ S2
(W \Q)w1 = W2\ Qw1
����
i=1 Qi ∈ S ′
� ∞�
i=1
Qi
�
w1
(A1 × A2)w1 =
=
∞�
i=1
∈S2
(Qi)w1
� �� �
� A2
∈ S2
∅
∈S2
∈ S2
∈ S2
w1 ∈ S1
�
= W2

S1 ⊗ S2
{A1 × A2 : Ai ∈ Si}
w2 ∈ W2
S1 ⊗ S2 ⊂ S ′
Q ∈ S1 ⊗ S2
m1,m2 B1,k ↑ W1 B2,k ↑ W2 m1(B1k) <
∞ m2(B2k) < ∞ k ∈ N Q ∈ S1 ⊗ S2
S1
S2
Qw1 ∈ S2 fQ
m2(W2) < ∞ :
W1 × W2 ∈ D
fQ : W1 → [0, ∞],w1 ↦→ m2(Qw1 )
gQ : W2 → [0, ∞],w2 ↦→ m1(Qw2 )
D := {B ∈ S1 ⊗ S2 : fB
fW1×W2 = m2((W1 × W2)w1) =m2(W2) =
B ∈ D B C ∈ D fB
Bi ∈ D
f P ∞
i=1
f B C = m2((B C )w1 )=m2((Bw1 )C )
m2(Bw 1 )

gQ
m2(W2) =∞
fQ
(A1 × A2) ∩ (B1 × B2) =(A1 ∩ B1) × (A2 ∩ B2) ∈ D
mi(Bi,k) < ∞
Ai ∈ Si
S1 ⊗ S2
Q ∈ S1 ⊗ S2
Def
= S({A1 × A2 : Ai ∈ Si})
= D({A1 × A2 : Ai ∈ Si}) ⊂ D
fQ S1
B2,k ↑ W2
∀k ∈ N : m2(B2k) < ∞
S2
m2,k : S2 → [0, ∞),A2 ↦→ m2(A2 ∩ B2,k)
fQ,k : W1 → [0, ∞],w1 ↦→ m2(Qw1
sup fQk
k∈N
= sup
k∈N
�
= m2
m2 (Qw1 ∩ B2,k)
∞�
Qw1 ∩ B2,k
k=1
∩ B2,k)
�
= m2(Qw1 ∩ W )=m2(Qw1 )
= fQ
(Wi,Si,mi),i =1, 2 Bi,k ↑ Wi
m S1 ⊗ S2
m(A1 × A2) =m1(A1)m2(A2)
�
Q ∈ S1 ⊗ S2
m(Q) = m2(Qw1 )dm1
�
= m1(Qw2)dm2
m = m1 ⊗ m2
Ck S1 ⊗ S2 Ck ↑ W1 × W2 m(Ck) < ∞

Q ↦→ m2(Qw1) ≥ 0 S1
�
m : S1 ⊗ S2 → [0, ∞],Q↦→ m2(Qw1)dm1
S1 ⊗ S2 :
�
Qi ∈ S1 ⊗ S2
m(Q) = m2(Qw1) dm1 ≥ 0
� �� �
�
≥0
m(∅) = m2(∅w1 )dm1
�
= 0dm1 =0
�
∞�
m
�
=
�
⎛�
∞�
⎝
� ⎞
�
⎠ dm1 =
i=1
Qi
=
=
m2
� ∞ �
i=1
∞�
m(Qi)
i=1
i=1
Qi
w1
m2(Qiw1 )dm1 =
∞�
�
i=1
m2
� ∞�
Qiw1
i=1
m2(Qiw1 )dm1
A1 × A2 ∈ S1 ⊗ S2
�
m(A1 × A2) = m2(A2)1A1dm1 = m1(A1)m2(A2)
� m2(Qw1 )dm1
m ∗ : S1 ⊗ S2 → [0, ∞],Q↦→
m = m ∗
Ck := B1,k × B2,k ↑ W1 × W2
�
m1(Qw2)dm2(w2)
m(Ck) =m(B1,k × B2,k) =m1(B1,k)m2(B2,k) < ∞
R p R
�
dm1

S1 ⊗ S2
�
l p ((a, b]) =
p�
(bi − ai) =
i=1
p�
l((ai,bi])
i=1
(Wi,Si,mi),i=1, 2 Bi,k ↑ Wi mi(Bi,k) < ∞
w2 ↦→
w1 ↦→
f : W1 × W2 → [0, ∞]
�
�
� ��
fd(m1 ⊗ m2) =
� ��
f(w1,w2)w2dm1
f(w1,w2)w1 dm2
fw2 dm1
� � ��
dm2 =
fw1 dm2
fwi mi wi
u = � n
i=1 ai1Ai ∈ T (W1 × W2)
uw2 =
n�
i=1
�
uw2dm1 dm2 =
=
s.o.
=
=
ai1Ai,w2
∈ T (W1)
�
dm1
� �� �
�n
ai1Ai,w2dm1 dm2
n�
ai
i=1
�
i=1
m1((Ai)w2 )dm2
n�
aim1 ⊗ m2(Ai)
i=1
�
udm1 ⊗ m2

� ��
un ∈ T (W1 × W2),un ↑ f un,w2 ∈ T (W1) un,w2 ↑ fw2
�
�
fdm1 ⊗ m2
Def
= lim undm1 ⊗ m2
n→∞
� ��
un∈T
= lim
n→∞
un,w2dm1 �
�
dm2
�� �
�
≥
�
↑
= lim
n→∞
un,w2
� �� �
dm1dm2
� ��
≥ ↑
�
=
� ��
lim un,w2dm1 n→∞
�
dm2
= fw2dm1 dm2
� ��
fw1 dm2
|fwi| = |f|wi
(fwi) + = (f + )wi
(fwi) − = (f − )wi
|fw1 |dm2
�
dm1 =
�
dm1 =
1.)
=
=
w1 ↦→
� ��
�
�
=
� ��
�
|fw1 |dm2
f + w1 dm2
|fw2 |dm1
�
dm2
|f|dm < ∞
�
f + dm1 ⊗ m2 −
�
fdm1 ⊗ m2
� ��
dm1 −
�
f − dm1 ⊗ m2
f − w1 dm2
�
dm1

∀t ∈ T : f(t, ·) ∈ L 1
limn→∞ tn = t0
x ∈ X
f(·,x):T → K t0 ∈ T
δ>0 g ≥ 0
∀t ∈ B(t0,δ): |f(t, ·)| ≤g
f : T × X → K
lim
n→∞ f(tn,x) =
�
f lim
n→∞ tn,x
�
∈ L 1
�
� �
lim
n→∞
f(tn,x)dm(x)
X
= f
X
lim
n→∞ tn,x
�
dm(x)
�
f(t, x)dm(x)
X
B(t0,δ)
t0
�
tn ∈ B(t0,δ) |f(tn, ·)| ≤g
lim
n→∞
X
f(tn,x)
� �� �
∈L1 �
dm(x)
|f(tn,·)|≤g
=
�
t0
=
X
X
lim
n→∞ f(tn,x)dm(x)
f(t0,x)dm(x)
I ⊂ R
∀t ∈ I : f(t, ·) ∈ L
t ∈ I f : I × X → K
1
∀x ∈ X
∂f
∂t (t0,x)
δ>0 g ≥ 0
�
�
�
∀t ∈ I ∩ B(t0,δ),t�= t0 : �
f(t, x) − f(t0,x) �
�
� t − t0
� ≤ g(x)
∃δ >0 ∀x ∈ X ∀t ∈ B(t0,δ): ∂f
(t, x) t
∂t

�
d
dt X
g ≥ 0
∀x ∈ X ∀t ∈ B(t0,δ):
� �
�
�
∂f �
� (t, x) �
∂t � ≤ g(x)
∂f
∂t (t0,x):X → K
�
F : I → K,t↦→ f(t, x)dm(x) t0
�
d
dt
X
�
�
f(t, x)dm(x) �
�
=
∂f
∂t (t0,x)dm(x)
X
B(t0,δ)
∂f
∂t (t0,x)
limn→∞ tn = t0
�
�
f(t, x)dm(x) �
� t=t0
� t=t0
X
t0 �= tn ∈ B(t0,δ)
�
Def
= lim
n→∞
|·|≤g
=
=
�
�
X
X
X
lim
n→∞
f(tn,x) − f(t0,x)
dm(x)
tn − t0
f(tn,x) − f(t0,x)
dm(x)
tn − t0
∂f
∂t (t0,x)dm(x)
∂f
∂t (t, x) ∀x ∈ X ∃sn,x ∈ B(t0,δ):
�
�
�
�
f(tn,x) − f(t0,x) �
�
�
� =
�
�
�
∂f
� ∂t (sn,x,x)
�
�
�
� ≤ g(x)
tn − t0
|t0 − sn| ≤|t0 − tn| f(tn,x),f(t0,x)
∂f
∂t
= lim
n→∞
f(tn,x) − f(t0,x)
tn − t0

�
d
f(t, x)
dt X � �� �
∈L1 dm(x)|t=t0 = lim
n→∞
�
X
X
f(tn,x) − f(t0,x)
dm(x)
tn − t0
�
b)
∂f
= lim
n→∞
X ∂t (sn,x,x)dm(x)
�
|·|≤g,c) ∂f
= lim
X
n→∞ ∂t (sn,x,x)dm(x)
�
∂f
=
∂t (t0,x)dm(x)

|A| := det A
(W1,S1,m) f → (W2,S2,mf ) h → (R, B)
h ≥ 0
mf
Ai ∈ S2
mf : S2 → R,A↦→ m[f −1 (A)]
h ◦ f
�
�
1 f −1 (Ai)(w1) =
�
1Ai
=
◦ fdm =
�
hdmf = (h ◦ f)dm
⇐⇒ h ◦ f
�
hdmf = (h ◦ f)dm
� 1 w1 ∈ f −1 (Ai)
0
�
1 f(w1) ∈ Ai
0
= 1Ai(f(w1))
= (1Ai◦f)(w1) �
Def
= mf (Ai) =
W1
W2
1 f −1 (Ai)dm = m(f −1 (Ai))
�
1Aidmf
mf

u = �n i=1 ai1Ai ∈ T (W2)
�
�n
ai1Ai dmf
n�
=
i=1
=
i=1
ai
�
� � n �
i=1
1Ai dmf =
ai1Ai
n�
ai
i=1
�
◦ fdm
h ∈ T ∗ un = � kn
i=1 ai,n1Ai,n ∈ T un ↑ h
un ◦ f ↑ h ◦ f
�
hdmf
l p
�
Def
=
�
lim
n→∞ un
�
2.)
= lim
n→∞
�
�
=
hdmf =
�
����
≥0,↑
�
dmf = lim
n→∞
un ◦ f dm
� �� �
≥0,↑
lim
n→∞ un ◦ fdm =
h + dmf =
h − dmf =
=
=
�
�
(h + ◦ f)dm
�
(h − ◦ f)dm
�
h ◦ fdm
1Ai
undmf
�
h + �
dmf − h − dmf
�
h + �
◦ fdm− h − ◦ fdm
�
h ◦ fdm
∀c ∈ R n , ∀A ∈ B p : l p (c + A) =l p (A)
l p (c +(a, b]) =
=
(a, b] B p
p�
(bi + ci − (ai + ci))
i=1
p�
(bi − ai) =l p ((a, b])
i=1
◦ fdm

Eij,−c
Eij,c,Pij,Mic
R p ∀A ∈ B p
l p (E −1
ij,c (A)) = lp (A)
l p (P −1
ij (A)) = lp (A)
∀c �= 0: l p (Mic(A)) = |c|l p (A)
(a, b] ∈ Rp Rp C :=
−c
⎧⎛
⎞
⎫
⎨ a1 + x1
⎬
⎝ ai + xi − c(aj + xj) ⎠ : xi ∈ (0,bi − ai]
⎩
⎭
= Eij,−c((a, b])
an + xn
E −1
ij,c = Eij,−c :(a, b] → C, y ↦→
E −1
ij,−c = Eij,c : C → (a, b],y ↦→
⎛
⎝
y1
⎞
yi − cyj ⎠
yn
⎛
⎝
y1
⎞
yi + cyj ⎠
yn
��
li Eij,c) −1 (a, b] � �
|w1×...×wi−1×wi+1×...×wp
= li((ai − c(aj + xj),bi − c(aj + xj)]) ·
= (bi − ai) ·
n�
j=1,j�=i
1 (aj,bj](wj)
n�
j=1,j�=i
1 (aj,bj](wj)

=
=
l p ((Eij,c) −1 (a, b])
�
li((Eij,c) −1 (a, b])|w1×...×wi−1×wi+1×...×wp
dl1 ⊗ ...⊗ dli−1 ⊗ dli+1 ⊗ ...⊗ dlp
�
p�
(bi − ai) 1 (aj,bj]
j=1,j�=i
dl1 ⊗ ...⊗ dli−1 ⊗ dli+1 ⊗ ...⊗ dlp
p�
�
= (bi−ai) 1 (aj,bj]dlj
=
p�
(bi − ai)
i=1
= l p ((a, b])
j=1,j�=i
Pij((a, b]) = (a1,b1] × ...× (aj,bj] × ...× (ai,bi] × ...× (an,bn]
l p (Pij((a, b])) =
p�
(bi − ai) =l p ((a, b])
i=1
Mi,c =(a1,b1] × ...× (cai,cbi] × ...× (an,bn]
l p (Mi,c((a, b])) = (b1 − a1) · ...·|c|·(bi − ai) · ...· (bn − an)
=
p�
|c| (bi − ai) =|c|·l p ((a, b])
i=1
p × p ∀A ∈ B p :
l p (B −1 (A)) = | det B −1 |·l p (A)
Ek
M1,c1 ···Mp,cp · E1 ···EnB = 1
M1,c1 ···Mp,cp · E1 ···En = B −1
Mi,ci

ci �= 0
| det B −1 | = | det M1,c1|···|det Mp,cp|
=
p�
|ci|
l
i=1
p (B −1 (A)) = l p (M1,c1 ···Mp,cp · E1 ···EnA)
= l p (E1 ···EnA) ·
= l p (A)
p�
|ci|
i=1
= | det B −1 |·l p (A)
p�
|ci|
h :[a−δ, b+δ] → R p ∀x, y ∈ (a, b)
i=1
� h(x) − h(y) �2≤� x − y �2 sup
c∈[0,1]
� Dh(x + c(y − x)) �2
g :(−δ, 1+δ) → R,t↦→ 〈h(x + t(y − x)),h(y) − h(x)〉
g
t ∈ [0, 1]
′ (t) =
� �
d
h(x + t(y − x)),h(y) − h(x)
dt
= 〈Dh(x + t(y − x)) · (y − x),h(y) − h(x)〉
= � Dh(x + t(y − x)) · (y − x) �� h(y) − h(x) �
≤ sup � Dh(x + c(y − x)) �2� y − x �2� h(y) − h(x) �2
0≤c≤1
∃s ∈ [0, 1] :
� h(y) − h(x) � 2 2 = 〈h(y) − h(x),h(y) − h(x)〉
Def
= g(1) − g(0) = g ′ (s)
≤ sup
0≤c≤1
� Dh(x + c(y − x)) �2
� y − x �2� h(y) − h(x) �2

∀A ⊂ U, A ∈ B p
U ⊂ R p [a, b] ⊂ R n [a, b] ⊂ U
dist([a, b],U C ) > 0
[a, b] U C
U, V ⊂ R p t : U → V t, t −1
l p �
(t(A)) =
f ≥ 0
�
fdl p �
=
V
U
A
| det Dt|dl p
(f ◦ t)| det Dt|dl p
f : V → R ⇐⇒ f ◦ t| det Dt| : U → R
Dt −1
�
V
[a, b] ⊂ U
fdl p �
=
U
f ◦ t| det Dt|dl p
∃δ1 > 0 ∀z ∈ [a, b] :[z − δ1,z+ δ1] ⊂ U
M := sup � (Dt(y))
y∈[a,b]
−1 �2< ∞
Dt δ
� x − y �2< δ⇒ sup
x,y∈[a,b]
� Dt(x) − Dt(y) �2≤ ε
d
(a, b] (a i ,b i ] i = 1,...,N
∃C >0: δ
C ≤ d √ p

∀x ∈ (a i ,b i ]
l p
i =1,...,n ci ∈ [a i ,b i ]
| det Dt(ci)| = min
y∈[ai ,bi | det Dt(y)|
]
Ti := Dt(ci)
h(x) := t(x) − Tix
y := ci
� t(x) − t(ci) − Ti(x − ci) �
≤ � x − ci �2 sup
s∈[0,1]
� Dt(x + s(ci − x)) − Ti �
Vor
≤ δε
t(x) =t(ci)+Dt(ci) (x − ci)+r(x) � r(x) �2≤ δε
� �� �
=Ti
t((a i ,b i ]) ⊂ t(ci) − Tici + Ti((a i ,b i ]) + (−2δε, 2δε]
δp ≤
Cp l p � t((a i ,b i ]) �
p�
|bj − aj| = l p � (a i ,b i ] �
j=1
≤ l p � Ti((a i ,b i ] � + l p � T −1
i Ti(−2δε, 2δε] �
= | det Ti|·l p � (a i ,b i ] � + | det Ti|·det |T −1
i | l
� �� �
≤M
p ((−2δε, 2δε])
� �� �
=(4εδ) p
≤ |det Ti|· � l p � (a i ,b i ] � + M(4εδ) p�
≤ |det Ti|·l p � (a i ,b i �
�
]
1+ M(4εδ)p
δ p C −p
�
| det Ti|1 (a i ,b i ] ≤|det Dt|1 (a i ,b i ]

ε>0
i =1,...,n
l p (t((a, b])) ≤ (1 + M(4εC) p )
n�
| det Ti|·l p ((a i ,b i ])
i=1
≤ (1 + M(4εC) p �
)
l p �
(t((a, b])) ≤
(a,b]
(a, b] B(U)
f ≥ 0
�
l p �
(t(A)) ≤
V
fdl p �
≤
U
A
(a,b]
| det Dt|dl p
| det Dt|dl p
f ◦ t| det Dt|dl p
| det Dt|dl p
B ∈ B,B ⊂ V A := t−1 (B)
1A(w) =
�
1
0
−1 w ∈ A = t (B)
w �∈ A = t−1 =
�
1
0
(B)
t(w) ∈ B
t(w) �∈ B
= 1B(t(w)) = (1B ◦ t)(w)
�
V
1Bdl p = l p (B) =l p (t(A))
�
≤ | det Dt|dl
A
p
�
= 1B ◦ t| det Dt|dl p
U

u = � k
i=1 ai1Bi ∈ T
�
V
un ↑ f
�
�
U
udl p =
fdl
V
p
≤
=
�
V
�
V
k�
i=1
�
U
k�
i=1
ai
�
ai1Bi dlp =
U
k�
i=1
ai
�
1Bi ◦ t| det Dt|dl p
u ◦ t| det Dt|dl p
un↑f
= lim
un◦t↑f◦t
=
n→∞
�
undl
V
p
V
1Bi dlp
≤ lim
n→∞
�
un ◦ t| det Dt|dl
U
p
�
lim
U
n→∞ un ◦ t| det Dt|dl p
=
�
f ◦ t| det Dt|dl p
f ≥ 0
fdl p �
=
U
U
f ◦ t| det Dt|dl p
t−1 : V → U f ◦ t| det Dt| f
f ◦ t| det Dt|dl p �
≤
V �
= fdl p
�
V
U
f + ,f− f
V
fdl p �
=
1 t(A)(w) =
=
f ◦ t ◦ t −1 | det Dt|| det Dt −1 |dl p
f ◦ t| det Dt|dl p
�
1 w ∈ t(A)
0 w �∈ t(A)
� 1 t −1 (w) ∈ A
0 t −1 (w) �∈ A
= (1A ◦ t −1 )(w)

l p (t(A)) =
p ≥ 2
4.)
=
=
�
�
�
V
1t(A)dl p �
=
(1A ◦ t
V
−1 )(w)dl p
1A ◦ t
U
−1 ◦ t| det Dt|dl p
A
| det Dt|dl p
t :(0, ∞) × (0,π) p−2 × (0, 2π) → R p
⎛
⎞
r cos a1
⎜
r sin a1 cos a2
⎟
⎜
⎟
⎜
⎟
⎜
⎟
(r, a1,...,ap−1) ↦→ ⎜ r sin a1 ...sin ak−1 cos ak ⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝ r sin a1 ...sin ap−2 cos ap−1
⎠
r sin a1 ...sin ap−2 sin ap−1
f :(Rp , Bp ) → (R, B) f ≥ 0
�
fdl p �
=
R p
(bn)n
(0,∞)×(0,π) p−2 ×(0,2π)
f(t(r, a1,...,ap−1))
r p−1 sin p−2 a1 ...sin ap−1dl p (r, a1,...,ap−1)
Hp := {y ∈ R p : yp−1 ≥ 0,yp =0}
Hp
∀n ∈ N : bn,p−1 ≥ 0
∀n ∈ N : bn,p = 0
bp−1 = lim
n→∞ bn,p−1 ≥ 0
bp = lim
n→∞ bn,p = 0

∈ Hp
Hp
l p (Hp) = l p
�
∞�
�
[−n, n] × ...× [−n, n] × [0,n] ×{0}
n=1
= lim
n→∞ lp ([−n, n] × ...× [−n, n] × [0,n] ×{0})
= lim
n→∞ 0=0
t :(0, ∞) × (0,π) p−2 × (0, 2π) → R p \Hp
t −1 : R p \Hp → (0, ∞) × (0,π) p−2 × (0, 2π)
⎛
r =� y �2
⎜ a1 =arccos
⎜
y ↦→ ⎜
⎝
y1
⎞
⎟
�y�2
⎟
y2
a2 =arccos
⎟
�y�2sin a1
⎟
⎠
ap−1 =arccos
yp−1
�y�2sin a1··· sin ap−2
=
det
�
Dt
�
� cos a1
�
� sin a1 cos a2
�
�
−r sin a1
r cos a1 cos a2
0
−r sin a1 sin a2
�
0 �
�
0 �
�
�
�
= cos 2 a1r p−1 sin p−2 �
�
�
�
a1 �
�
�
cos a2
sin a2 cos a3
− sin a2
cos a2 cos a3
0
− sin a2 sin a3
�
�
�
�
�
�
�
−(−r sin a1)r p−2 sin p−1 �
�
�
�
a1 �
�
�
cos a2
sin a2 cos a3
− sin a2
cos a2 cos a3
0
− sin a2 sin a3
�
�
�
�
�
�
�
= r p−1 sin p−2 �
�
�
�
a1 �
�
�
cos a2
sin a2 cos a3
− sin a2
cos a2 cos a3
0
− sin a2 sin a3
�
0 �
�
0 �
�
�
�

det Dt = r p−1 sin p−2 a1 sin p−3 a2 ···sin 2 ap−3 sin ap−2
l p (H p )=0
�
R p
fdl p =
�
�
R p
fdl p �
=
Rp \Hp fdl p
(0,∞)×(0,π) p−2 ×(0,2π)
(f ◦ t)r p−1 ·
· sin p−2 a1 ...sin ap−1dl p (r, a1,...,ap−1)

200 410
195 95
195 32
71 44
365 204
376 188
390 188
64 153
142 142
29 179
Q 18 142
50, 195 342
283 128
246 383
271 162
74, 215 81
341 393
148 301
380 81
123, 242 264
282 N 1
282 89
282 199
64 220
142 261
422 338
35, 232 12
Z 8 441
62 330
331 178
35, 195 360
44 R 51
43 111
298 134, 324
4 299, 363

360
379
183, 288, 294
246
183, 288, 294
108
57, 196
41
87, 401
71
64
112, 236
89
138
136
267
285
196
66
C 226

f : V → W
a ◦ (b ◦ c) =(a ◦ b) ◦ c
f : V → V
⎛
0
⎜ 1
⎜
⎝ 0
−c0
−c1
⎞
⎟
⎠
0 1 −cn−1
� � n
k
S( )
f : V → V
f : V → W

A = A T
1A
f : V → W
C
f : V → W
{A : A ⊂ W }
f,f −1

σ −
σ −
σ −
A T