7. Consequences of the Hahn-Banach theorems - Aarhus Universitet

7. Consequences of the Hahn-Banach theorems
Klaus Thomsen matkt@imf.au.dk
Institut for Matematiske Fag
Det Naturvidenskabelige Fakultet
Aarhus Universitet
September 2005
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

We read in W. Rudin: Functional Analysis
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

We read in W. Rudin: Functional Analysis
Covering Chapter 3, from page 59 up to 63.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(The corollary on p. 59) Let X be a locally convex topological
vector space. Then X ∗ separates the points of X, i.e. when x = y
in X there is a l ∈ X ∗ such that l(x) = l(y).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(The corollary on p. 59) Let X be a locally convex topological
vector space. Then X ∗ separates the points of X, i.e. when x = y
in X there is a l ∈ X ∗ such that l(x) = l(y).
Proof.
Apply the Hahn-Banach separation theorem, Theorem 3.4 b), with
A = {x} and B = {y}.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.5) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If x0 ∈ X is NOT in the closure M there is
a functional l ∈ X ∗ such that l(M) = {0} and l (x0) = 1.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.5) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If x0 ∈ X is NOT in the closure M there is
a functional l ∈ X ∗ such that l(M) = {0} and l (x0) = 1.
Proof.
Apply the Hahn-Banach separation theorem, Theorem 3.4 b), with
A = M and B = {x0} to obtain L ∈ X ∗ and γ ∈ R such that
Re L(M) < γ < Re L(x0).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.5) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If x0 ∈ X is NOT in the closure M there is
a functional l ∈ X ∗ such that l(M) = {0} and l (x0) = 1.
Proof.
Apply the Hahn-Banach separation theorem, Theorem 3.4 b), with
A = M and B = {x0} to obtain L ∈ X ∗ and γ ∈ R such that
Re L(M) < γ < Re L(x0).
Since 0 ∈ M we conclude that γ > 0. Set l = L(x0) −1 L.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.6) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If f : M → Φ is a continuous linear
functional on M, there is an l ∈ X ∗ such that l|M = f .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.6) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If f : M → Φ is a continuous linear
functional on M, there is an l ∈ X ∗ such that l|M = f .
Proof.
We may assume that f is not identically 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.6) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If f : M → Φ is a continuous linear
functional on M, there is an l ∈ X ∗ such that l|M = f .
Proof.
We may assume that f is not identically 0.
Set
M0 = {x ∈ M : f (x) = 0}.
and choose x0 ∈ M such that f (x0) = 1.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.6) Let X be a locally convex topological vector space
and M ⊆ X a subspace. If f : M → Φ is a continuous linear
functional on M, there is an l ∈ X ∗ such that l|M = f .
Proof.
We may assume that f is not identically 0.
Set
M0 = {x ∈ M : f (x) = 0}.
and choose x0 ∈ M such that f (x0) = 1.
Then x0 /∈ M0: See the next slide!
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Remember that M is equipped with the relative topology inherited
from X. Since f : M → Φ is continuous with respect to this
topology, M0 = f −1 ({0}) is closed in that topology.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Remember that M is equipped with the relative topology inherited
from X. Since f : M → Φ is continuous with respect to this
topology, M0 = f −1 ({0}) is closed in that topology.
This means that M\M0 = M ∩ W for some open set W of X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Remember that M is equipped with the relative topology inherited
from X. Since f : M → Φ is continuous with respect to this
topology, M0 = f −1 ({0}) is closed in that topology.
This means that M\M0 = M ∩ W for some open set W of X.
Then
W ∩ M0 = W ∩ M ∩ M0 = (M\M0) ∩ M0 = ∅.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Remember that M is equipped with the relative topology inherited
from X. Since f : M → Φ is continuous with respect to this
topology, M0 = f −1 ({0}) is closed in that topology.
This means that M\M0 = M ∩ W for some open set W of X.
Then
W ∩ M0 = W ∩ M ∩ M0 = (M\M0) ∩ M0 = ∅.
This means that M0 ⊆ W c and it follows that M0 ⊆ W c .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Remember that M is equipped with the relative topology inherited
from X. Since f : M → Φ is continuous with respect to this
topology, M0 = f −1 ({0}) is closed in that topology.
This means that M\M0 = M ∩ W for some open set W of X.
Then
W ∩ M0 = W ∩ M ∩ M0 = (M\M0) ∩ M0 = ∅.
This means that M0 ⊆ W c and it follows that M0 ⊆ W c .
Since x0 ∈ M\M0 = M ∩ W, we conclude that x0 /∈ M, as asserted.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
By Theorem 3.5 there is a l ∈ X ∗ such that l(M0) = {0} and
l(x0) = 1.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
By Theorem 3.5 there is a l ∈ X ∗ such that l(M0) = {0} and
l(x0) = 1.
Let x ∈ M. Then x − f (x)x0 ∈ M0 (since
f (−f (x)x0) = f (x) − f (x)f (x0) = 0), and hence
l(x − f (x)x0) = 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
By Theorem 3.5 there is a l ∈ X ∗ such that l(M0) = {0} and
l(x0) = 1.
Let x ∈ M. Then x − f (x)x0 ∈ M0 (since
f (−f (x)x0) = f (x) − f (x)f (x0) = 0), and hence
l(x − f (x)x0) = 0.
It follows that l(x) = f (x).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.7) Let X be a locally convex topological vector space.
Let B ⊆ X be a convex, balanced and closed subset. Let
x0 ∈ X \B. There is an l ∈ X ∗ such that |l(x)| ≤ 1 for all x ∈ B
while l(x0) > 1.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.7) Let X be a locally convex topological vector space.
Let B ⊆ X be a convex, balanced and closed subset. Let
x0 ∈ X \B. There is an l ∈ X ∗ such that |l(x)| ≤ 1 for all x ∈ B
while l(x0) > 1.
Proof.
It follows from Theorem 3.4 b) that there are a Λ0 ∈ X ∗ and a real
number γ ∈ R such that
for all b ∈ B.
ReΛ0(x0) < γ < ReΛ0(b)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Theorem
(Theorem 3.7) Let X be a locally convex topological vector space.
Let B ⊆ X be a convex, balanced and closed subset. Let
x0 ∈ X \B. There is an l ∈ X ∗ such that |l(x)| ≤ 1 for all x ∈ B
while l(x0) > 1.
Proof.
It follows from Theorem 3.4 b) that there are a Λ0 ∈ X ∗ and a real
number γ ∈ R such that
ReΛ0(x0) < γ < ReΛ0(b)
for all b ∈ B.
Since 0 ∈ B we see that γ < 0. Set Λ00 = 1
γ Λ0, and note that
Re Λ00(x0) > 1 > Re Λ00(b), b ∈ B. (1)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Because Λ00 is linear, it follows easily that Λ00(B) is convex and
balanced since B is. It is then easy to see (!!) that there is an
r ≥ 0 such that
{z ∈ C : |z| < r} ⊆ Λ00(B) ⊆ {z ∈ C : |z| ≤ r}.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Because Λ00 is linear, it follows easily that Λ00(B) is convex and
balanced since B is. It is then easy to see (!!) that there is an
r ≥ 0 such that
{z ∈ C : |z| < r} ⊆ Λ00(B) ⊆ {z ∈ C : |z| ≤ r}.
Note that r ≤ 1 by (1), and that
for all b ∈ B.
|Λ00(b)| ≤ r ≤ 1 < ReΛ00(x0) (2)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Consequences of the Hahn-Banach theorems
Proof.
Because Λ00 is linear, it follows easily that Λ00(B) is convex and
balanced since B is. It is then easy to see (!!) that there is an
r ≥ 0 such that
{z ∈ C : |z| < r} ⊆ Λ00(B) ⊆ {z ∈ C : |z| ≤ r}.
Note that r ≤ 1 by (1), and that
|Λ00(b)| ≤ r ≤ 1 < ReΛ00(x0) (2)
for all b ∈ B.
Choose α ∈ C such that |α| = 1 and αΛ00(x0) = |Λ00(x0)|. Set
Λ = αΛ00, note that
for all b ∈ B.
Λ(x0) ≥ ReΛ00(x0) > 1 ≥ |Λ00(b)| = |Λ(b)|
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Let X be a real or complex vector space, and M a collection of
lineat functionals X → Φ.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Let X be a real or complex vector space, and M a collection of
lineat functionals X → Φ.
The sets of the form
{y ∈ X : |l(y) − l(x)| < ǫ} (3)
where x ∈ X,ǫ > 0 and l ∈ M vary, is a subbase for a topology on
X,
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Let X be a real or complex vector space, and M a collection of
lineat functionals X → Φ.
The sets of the form
{y ∈ X : |l(y) − l(x)| < ǫ} (3)
where x ∈ X,ǫ > 0 and l ∈ M vary, is a subbase for a topology on
X,
namely the topology where a subset of X is open if and only if it is
the union of sets which are the intersection of a finite collection of
such sets.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Let X be a real or complex vector space, and M a collection of
lineat functionals X → Φ.
The sets of the form
{y ∈ X : |l(y) − l(x)| < ǫ} (3)
where x ∈ X,ǫ > 0 and l ∈ M vary, is a subbase for a topology on
X,
namely the topology where a subset of X is open if and only if it is
the union of sets which are the intersection of a finite collection of
such sets.
This will be called the M-topology of X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
We may assume that U and V are intersections of finite collections
of sets of the form (3).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
We may assume that U and V are intersections of finite collections
of sets of the form (3).
It follows that there is a set of the form (3) which contains x0 but
not y0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
We may assume that U and V are intersections of finite collections
of sets of the form (3).
It follows that there is a set of the form (3) which contains x0 but
not y0.
I.e.
x0 ∈ {y ∈ X : |l(y) − l(x)| < ǫ}
while |l(y0) − l(x)| ≥ ǫ for some x ∈ X,l ∈ M and some ǫ > 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Lemma
The M-topology is Hausdorff if and only if M separates the points
of X.
Proof.
Let x0,y0 ∈ X, x0 = y0. If the M-topology is Hausdorff there are
open set U,V such that x0 ∈ U,y0 ∈ V and U ∩ V = ∅.
We may assume that U and V are intersections of finite collections
of sets of the form (3).
It follows that there is a set of the form (3) which contains x0 but
not y0.
I.e.
x0 ∈ {y ∈ X : |l(y) − l(x)| < ǫ}
while |l(y0) − l(x)| ≥ ǫ for some x ∈ X,l ∈ M and some ǫ > 0.
Then l(x0) = l(y0), and we conclude that M separates the points
of X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, assume that M separates the points of X. Let
x0,y0 ∈ X, x0 = y0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, assume that M separates the points of X. Let
x0,y0 ∈ X, x0 = y0.
There is then a functional l ∈ M such that l(x0) = l(y0).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, assume that M separates the points of X. Let
x0,y0 ∈ X, x0 = y0.
There is then a functional l ∈ M such that l(x0) = l(y0).
Set ǫ = 1/2 |l(x0) − l(y0)| > 0, and note that
x0 ∈ {y ∈ X : |l(y) − l(x0)| < ǫ} ,
y0 ∈ {y ∈ X : |l(y) − l(y0)| < ǫ} .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, assume that M separates the points of X. Let
x0,y0 ∈ X, x0 = y0.
There is then a functional l ∈ M such that l(x0) = l(y0).
Set ǫ = 1/2 |l(x0) − l(y0)| > 0, and note that
Since
x0 ∈ {y ∈ X : |l(y) − l(x0)| < ǫ} ,
y0 ∈ {y ∈ X : |l(y) − l(y0)| < ǫ} .
{y ∈ X : |l(y) − l(y0)| < ǫ} ∩ {y ∈ X : |l(y) − l(x0)| < ǫ} = ∅,
the proof is complete.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Lemma
Let l1,l2,... ,ln and l be linear functional on the vector space X.
The following are equivalent:
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Lemma
Let l1,l2,... ,ln and l be linear functional on the vector space X.
The following are equivalent:
(a) There are scalars α1,α2,... ,αn such that
l = α1l1 + α2l2 + · · · + αnln.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Lemma
Let l1,l2,... ,ln and l be linear functional on the vector space X.
The following are equivalent:
(a) There are scalars α1,α2,... ,αn such that
l = α1l1 + α2l2 + · · · + αnln.
(b) There is a γ > 0 such that |l(x)| ≤ γ max1≤i≤n |li(x)| for all
x ∈ X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Theorem
Theorem 3.10. Let X be a vector space and X ′ a separating vector
space of linear functional on X. Then the X ′ -topology makes X a
locally convex topological vector space whose dual is X ′ .
The proof is based on the following lemma of independent interest:
Lemma
Let l1,l2,... ,ln and l be linear functional on the vector space X.
The following are equivalent:
(a) There are scalars α1,α2,... ,αn such that
l = α1l1 + α2l2 + · · · + αnln.
(b) There is a γ > 0 such that |l(x)| ≤ γ max1≤i≤n |li(x)| for all
x ∈ X.
(c) For all x ∈ X, l1(x) = l2(x) = · · · = ln(x) = 0 ⇒ l(x) = 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(of lemma): (a) ⇒ (b):
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(of lemma): (a) ⇒ (b):
Set γ = n max1≤i≤n |αi|. Then
|l(x)| ≤ |α1| |l1(x)| + |α2| |l2(x)| + · · · + |αn| |ln(x)|
≤ n max
1≤i≤n |αi| max
1≤i≤n |li(x)| = γ max
1≤i≤n |li(x)| .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(of lemma): (a) ⇒ (b):
Set γ = n max1≤i≤n |αi|. Then
|l(x)| ≤ |α1| |l1(x)| + |α2| |l2(x)| + · · · + |αn| |ln(x)|
≤ n max
1≤i≤n |αi| max
1≤i≤n |li(x)| = γ max
1≤i≤n |li(x)| .
(b) ⇒ (c): This is trivial!
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).
Then F = l0 ◦ P : Φ n → Φ is a linear map such that
F ◦ π(x) = l0 ◦ P ◦ π(x) = l0 (π(x)) = l(x).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).
Then F = l0 ◦ P : Φ n → Φ is a linear map such that
F ◦ π(x) = l0 ◦ P ◦ π(x) = l0 (π(x)) = l(x).
Recall from linear algebra that the linear map F : Φ n → Φ is of the
form
F(x1,x2,... ,xn) = α1x1 + α2x2 + · · · + αnxn.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
(c) ⇒ (a): Define π : X → Φ n such that
π(x) = (l1(x),l2(x),... ,ln(x)).
It follows from (c) that we can define l0 : π(X) → Φ such that
l0 (π(x)) = l(x).
Let P : Φ n → π(X) be a linear projection such that P (Φ n ) = π(X).
Then F = l0 ◦ P : Φ n → Φ is a linear map such that
F ◦ π(x) = l0 ◦ P ◦ π(x) = l0 (π(x)) = l(x).
Recall from linear algebra that the linear map F : Φ n → Φ is of the
form
F(x1,x2,... ,xn) = α1x1 + α2x2 + · · · + αnxn.
It follows that
l(x) = F ◦ π(x) = F (l1(x),l2(x),... ,ln(x))
= α1l1(x) + α2l2(x) + · · · + αnln(x), x ∈ X.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Proof of Theorem 3.10. Almost by definition, the open sets in
X ′ -topology are the sets that are unions of sets of the form
l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn),
where each li ∈ X ′ and Vi is an open subset of the scalars Φ.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Proof of Theorem 3.10. Almost by definition, the open sets in
X ′ -topology are the sets that are unions of sets of the form
l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn),
where each li ∈ X ′ and Vi is an open subset of the scalars Φ.
To show that X is a topological vector space in the X ′ -topology we
check first that the vector space operations are continuous.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Proof of Theorem 3.10. Almost by definition, the open sets in
X ′ -topology are the sets that are unions of sets of the form
l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn),
where each li ∈ X ′ and Vi is an open subset of the scalars Φ.
To show that X is a topological vector space in the X ′ -topology we
check first that the vector space operations are continuous.
To show that addition is continuous, let U be an X ′ -open set. We
must show that
is X ′ -open in X × X.
R = {(x,y) ∈ X × X : x + y ∈ U} (4)
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Let (x ′ ,y ′ ) be an element of R. It suffices to show that there are
X ′ -open sets S,T such that (x ′ ,y ′ ) ∈ S × T ⊆ R. To this end
note that there are pairs (li,Vi),i = 1,2,... ,n, such that li ∈ X ′ ,
Vi ⊆ Φ is open and
x ′ + y ′ ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Let (x ′ ,y ′ ) be an element of R. It suffices to show that there are
X ′ -open sets S,T such that (x ′ ,y ′ ) ∈ S × T ⊆ R. To this end
note that there are pairs (li,Vi),i = 1,2,... ,n, such that li ∈ X ′ ,
Vi ⊆ Φ is open and
x ′ + y ′ ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U.
In particular, li(x ′ ) + li(y ′ ) ∈ Vi,i = 1,2,... ,n. Since the vector
space operations are continuous on Φ, there are open sets Si,Ti in
Φ such that li(x ′ ) ∈ Si,li(y ′ ) ∈ Ti and Si + Ti ⊆ Vi,i = 1,2,... ,n.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Let (x ′ ,y ′ ) be an element of R. It suffices to show that there are
X ′ -open sets S,T such that (x ′ ,y ′ ) ∈ S × T ⊆ R. To this end
note that there are pairs (li,Vi),i = 1,2,... ,n, such that li ∈ X ′ ,
Vi ⊆ Φ is open and
x ′ + y ′ ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U.
In particular, li(x ′ ) + li(y ′ ) ∈ Vi,i = 1,2,... ,n. Since the vector
space operations are continuous on Φ, there are open sets Si,Ti in
Φ such that li(x ′ ) ∈ Si,li(y ′ ) ∈ Ti and Si + Ti ⊆ Vi,i = 1,2,... ,n.
Set
and
S = l −1
1 (S1) ∩ l −2
2 (S2) ∩ · · · ∩ l −1
n (Sn)
T = l −1
1 (T1) ∩ l −2
2 (T2) ∩ · · · ∩ l −1
n (Tn).
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Then S,T are X ′ -open and (x ′ ,y ′ ) ∈ S × T since li(x ′ ) ∈ Si and
li(y ′ ) ∈ Ti,i = 1,2,... ,n.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Then S,T are X ′ -open and (x ′ ,y ′ ) ∈ S × T since li(x ′ ) ∈ Si and
li(y ′ ) ∈ Ti,i = 1,2,... ,n.
Furthermore, if s ∈ S,t ∈ T, we see that
li(s + t) = li(s) + li(t) ∈ Si + Ti ⊆ Vi,
proving that S + T ⊆ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U. I.e.
S × T ⊆ R.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Then S,T are X ′ -open and (x ′ ,y ′ ) ∈ S × T since li(x ′ ) ∈ Si and
li(y ′ ) ∈ Ti,i = 1,2,... ,n.
Furthermore, if s ∈ S,t ∈ T, we see that
li(s + t) = li(s) + li(t) ∈ Si + Ti ⊆ Vi,
proving that S + T ⊆ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U. I.e.
S × T ⊆ R.
The proof that scalar multiplication is also continuous is left to the
reader. See ’Noter og kommentarer’.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
In order to conclude that τ ′ makes X into a topological vector
space it remains now only to show that every one-point set {x} in
X is closed. This goes as follows:
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
In order to conclude that τ ′ makes X into a topological vector
space it remains now only to show that every one-point set {x} in
X is closed. This goes as follows:
For each y = x there is an element ly ∈ X ′ such that ly(x) = ly(y).
This is because X ′ separates points by assumption. It follows that
{x} c =
{z ∈ X : ly(z) = ly(x)} .
y∈{x} c
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
In order to conclude that τ ′ makes X into a topological vector
space it remains now only to show that every one-point set {x} in
X is closed. This goes as follows:
For each y = x there is an element ly ∈ X ′ such that ly(x) = ly(y).
This is because X ′ separates points by assumption. It follows that
{x} c =
{z ∈ X : ly(z) = ly(x)} .
Since
y∈{x} c
{z ∈ X : ly(z) = ly(x)} = l −1
y (Φ\{ly(x)}) ,
we see that {x} c is X ′ -open.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.
There are then pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.
There are then pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 = li(0) ∈ Vi for all i. Since Φ is a locally convex
topological space, there are open and convex neighborhoods Wi of
0 in Φ such that Wi ⊆ Vi.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.
There are then pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 = li(0) ∈ Vi for all i. Since Φ is a locally convex
topological space, there are open and convex neighborhoods Wi of
0 in Φ such that Wi ⊆ Vi.
Since the li’s are linear, l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) is
convex.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
X is locally convex: Consider an X ′ -open set U such that 0 ∈ U.
There are then pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 = li(0) ∈ Vi for all i. Since Φ is a locally convex
topological space, there are open and convex neighborhoods Wi of
0 in Φ such that Wi ⊆ Vi.
Since the li’s are linear, l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) is
convex.
Since l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) is X ′ -open and
0 ∈ l −1
1 (W1) ∩ l −2
2 (W2) ∩ · · · ∩ l −1
n (Wn) ⊆ U,
we have shown that X is locally convex in the X ′ -topology.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
To prove that X ∗ = X ′ , note first that every element of X ′ is
continuous with respect to the X ′ -topology, proving that X ′ ⊆ X ∗ .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
To prove that X ∗ = X ′ , note first that every element of X ′ is
continuous with respect to the X ′ -topology, proving that X ′ ⊆ X ∗ .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, let Λ ∈ X ∗ . Then Λ −1 ({z ∈ Φ : |z| < 1}) is X ′ open
and contains 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, let Λ ∈ X ∗ . Then Λ −1 ({z ∈ Φ : |z| < 1}) is X ′ open
and contains 0.
There are therefore pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, let Λ ∈ X ∗ . Then Λ −1 ({z ∈ Φ : |z| < 1}) is X ′ open
and contains 0.
There are therefore pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 ∈ Vi for all i. It follows then that Λ(x) = 0 when
x ∈ ker l1 ∩ ker l2 ∩ · · · ∩ ker ln. Indeed, if li(x) = 0 for all i, we see
that
tx ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U
for all t ∈ Φ, which implies that |t||Λ(x)| = |Λ(tx)| < 1 for all
t ∈ Φ. This is only possible when Λ(x) = 0.
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems

Weak vector space topologies
Proof.
Conversely, let Λ ∈ X ∗ . Then Λ −1 ({z ∈ Φ : |z| < 1}) is X ′ open
and contains 0.
There are therefore pairs (li,Vi),i = 1,2,... ,n, such that
0 ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U,
where li ∈ X ′ and the Vi’s are open in Φ.
Note that 0 ∈ Vi for all i. It follows then that Λ(x) = 0 when
x ∈ ker l1 ∩ ker l2 ∩ · · · ∩ ker ln. Indeed, if li(x) = 0 for all i, we see
that
tx ∈ l −1
1 (V1) ∩ l −2
2 (V2) ∩ · · · ∩ l −1
n (Vn) ⊆ U
for all t ∈ Φ, which implies that |t||Λ(x)| = |Λ(tx)| < 1 for all
t ∈ Φ. This is only possible when Λ(x) = 0.
Now Lemma 3.9 implies that
Λ ∈ Span{l1,l2,...,ln} ⊆ X ′ .
Klaus Thomsen 7. Consequences of the Hahn-Banach theorems