2. Topological vector spaces - Aarhus Universitet

2. Topological vector spaces
Klaus Thomsen matkt@imf.au.dk
Institut for Matematiske Fag
Det Naturvidenskabelige Fakultet
Aarhus Universitet
September 2005
Klaus Thomsen 2. Topological vector spaces

We read in W. Rudin: Functional Analysis
Klaus Thomsen 2. Topological vector spaces

We read in W. Rudin: Functional Analysis
Chapter 1, from Definition 1.4 to Theorem 1.12.
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
A base B for the topology of S is a collection of open sets in S
such that every open set is the union of sets from B.
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
A base B for the topology of S is a collection of open sets in S
such that every open set is the union of sets from B.
Example: In a metric space (X,d) the open balls
{y ∈ X : d(y,x) < ǫ} , x ∈ X, ǫ > 0,
is a base for the topology defined from the metric d.
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
A base B for the topology of S is a collection of open sets in S
such that every open set is the union of sets from B.
Example: In a metric space (X,d) the open balls
{y ∈ X : d(y,x) < ǫ} , x ∈ X, ǫ > 0,
is a base for the topology defined from the metric d.
Let X and Y be topological spaces. The product topology of the
product space X × Y is the topology with a base consisting of ’the
open rectangles’,
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
A base B for the topology of S is a collection of open sets in S
such that every open set is the union of sets from B.
Example: In a metric space (X,d) the open balls
{y ∈ X : d(y,x) < ǫ} , x ∈ X, ǫ > 0,
is a base for the topology defined from the metric d.
Let X and Y be topological spaces. The product topology of the
product space X × Y is the topology with a base consisting of ’the
open rectangles’,
i.e. the sets U × V = {(x,y) ∈ X × Y : x ∈ U, y ∈ V }, where
U ⊆ X and V ⊆ Y are open in X and Y , respectively.
Klaus Thomsen 2. Topological vector spaces

Bases and the product topology
Let S be a topological space with open sets τ.
A base B for the topology of S is a collection of open sets in S
such that every open set is the union of sets from B.
Example: In a metric space (X,d) the open balls
{y ∈ X : d(y,x) < ǫ} , x ∈ X, ǫ > 0,
is a base for the topology defined from the metric d.
Let X and Y be topological spaces. The product topology of the
product space X × Y is the topology with a base consisting of ’the
open rectangles’,
i.e. the sets U × V = {(x,y) ∈ X × Y : x ∈ U, y ∈ V }, where
U ⊆ X and V ⊆ Y are open in X and Y , respectively.
Exercise: Show that the product topology is a topology!
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Let Φ be either the real numbers R or the complex numbers C. Let
X be a vector space over Φ,
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Let Φ be either the real numbers R or the complex numbers C. Let
X be a vector space over Φ,
i.e. there are maps X × X ∋ (x,y) ↦→ x + y and
Φ × X ∋ (λ,x) ↦→ λx ∈ X such that the usual conditions
((x + y) + z = x + (y + z) etc. ) are all satisfied.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Let Φ be either the real numbers R or the complex numbers C. Let
X be a vector space over Φ,
i.e. there are maps X × X ∋ (x,y) ↦→ x + y and
Φ × X ∋ (λ,x) ↦→ λx ∈ X such that the usual conditions
((x + y) + z = x + (y + z) etc. ) are all satisfied.
Topological vector spaces
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Let Φ be either the real numbers R or the complex numbers C. Let
X be a vector space over Φ,
i.e. there are maps X × X ∋ (x,y) ↦→ x + y and
Φ × X ∋ (λ,x) ↦→ λx ∈ X such that the usual conditions
((x + y) + z = x + (y + z) etc. ) are all satisfied.
Topological vector spaces
The vector space X is a topological vector space when it has a
topology τ such that
(a) every point of X is a closed set, and
(b) the vector space operations are continuous.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Let Φ be either the real numbers R or the complex numbers C. Let
X be a vector space over Φ,
i.e. there are maps X × X ∋ (x,y) ↦→ x + y and
Φ × X ∋ (λ,x) ↦→ λx ∈ X such that the usual conditions
((x + y) + z = x + (y + z) etc. ) are all satisfied.
Topological vector spaces
The vector space X is a topological vector space when it has a
topology τ such that
(a) every point of X is a closed set, and
(b) the vector space operations are continuous.
NOTE : (b) means that the maps X × X ∋ (x,y) ↦→ x + y and
Φ × X ∋ (λ,x) ↦→ λx ∈ X are continuous when X × X and Φ × X
are equipped with the product topology.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Thus when x,y ∈ X and V ∈ τ is an open neighborhood V of
x + y there must be open neighborhoods Vx,Vy ∈ τ of x and y,
respectively, such that
Vx + Vy ⊆ V.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - the definition
Thus when x,y ∈ X and V ∈ τ is an open neighborhood V of
x + y there must be open neighborhoods Vx,Vy ∈ τ of x and y,
respectively, such that
Vx + Vy ⊆ V.
Similarly, when λ ∈ Φ,x ∈ X, and W ∈ τ is an open neighborhood
of λx there should be open neighborhoods Uλ of λ in Φ and
Vx ∈ τ of x in X such that
UλVx ⊆ W.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
R n
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
R n
C n
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
R n
C n
C[0,1] in the topology given by the metric
d(f ,g) = sup t∈[0,1] |f (t) − g(t)|
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
R n
C n
C[0,1] in the topology given by the metric
d(f ,g) = sup t∈[0,1] |f (t) − g(t)|
L2 [0,1] in
the topology given by the metric
1
d(f ,g) = 0 |f (t)|2 dt
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
The following are all examples of topological vector spaces:
R n
C n
C[0,1] in the topology given by the metric
d(f ,g) = sup t∈[0,1] |f (t) − g(t)|
L2 [0,1] in
the topology given by the metric
1
d(f ,g) = 0 |f (t)|2 dt
Do you know more examples?
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
Exercise: Let X be the vector space of convergent real sequences.
Show that there is a metric d defined on X such that
d ((xi) ∞ ∞
i=0 ,(yi) i=0
) = sup |xi − yi|,
and verify that X is a topological vector space with the topology τ1
defined by d.
i∈N
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
Exercise: Let X be the vector space of convergent real sequences.
Show that there is a metric d defined on X such that
d ((xi) ∞ ∞
i=0 ,(yi) i=0
) = sup |xi − yi|,
and verify that X is a topological vector space with the topology τ1
defined by d.
Define another topology τ2 on X such that sets of the following
form is a base:
i∈N
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
Exercise: Let X be the vector space of convergent real sequences.
Show that there is a metric d defined on X such that
d ((xi) ∞ ∞
i=0 ,(yi) i=0
) = sup |xi − yi|,
and verify that X is a topological vector space with the topology τ1
defined by d.
Define another topology τ2 on X such that sets of the following
form is a base:
For F ⊆ N a finite set, ǫ > 0 a positive number, and x = (xi) ∞
i=0 an
element of X, the set
is an element of the base.
i∈N
{(yi) ∞
i=1 : |xi − yi| < ǫ, i ∈ F }
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
Exercise: Let X be the vector space of convergent real sequences.
Show that there is a metric d defined on X such that
d ((xi) ∞ ∞
i=0 ,(yi) i=0
) = sup |xi − yi|,
and verify that X is a topological vector space with the topology τ1
defined by d.
Define another topology τ2 on X such that sets of the following
form is a base:
For F ⊆ N a finite set, ǫ > 0 a positive number, and x = (xi) ∞
i=0 an
element of X, the set
is an element of the base.
i∈N
{(yi) ∞
i=1 : |xi − yi| < ǫ, i ∈ F }
Show that X is a topological vector space in the τ2-topology and
that the τ2-topology is not the same as the τ1-topology.
Klaus Thomsen 2. Topological vector spaces

Topological vector spaces - examples
(Possible approach to the last assertion: For each n ∈ N, let
x n ∈ X be the sequence 0,0,0,... ,0,1,1,1,1,... . Then 0 is in
the τ2-closure of {x n : n = 1,2,3,... }, but not in the τ1-closure.)
Klaus Thomsen 2. Topological vector spaces