Locally convex topological vector space

In functional analysis and related areas of mathematics locally convex topological vector spaces or locally convex spaces are generalizations of semi normed spaces. Although such spaces are not necessarily normable they have, as with semi normed spaces, a convex local basis for 0. This condition is strong enough for the Hahn-Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces which are metrisable and complete with respect to this metric. They are generalizations of Banach spaces, which are complete vector spaces with respect to a norm.

Contents

1 Definition 2 Examples 2.1 Smooth functions 2.2 Continuous functions 3 Constructing a locally convex basis 4 Continuous linear mappings 5 Properties

Definition

A topological vector space <math>(X,\tau)<math> over the topological field <math>\mathbb{K}<math> is called locally convex topological vector space if there exists a neighbourhood basis (also called local basis) <math>\mathcal{B}<math> for the zero vector with the following properties:

For all <math>U<math> and <math>V<math> in <math>\mathcal{B}<math>

• <math>\exists W \in \mathcal{B}<math> with <math>W \subseteq U \cap V<math>
• <math>\forall \alpha \in \mathbb{K} \setminus \{0\} : \alpha U \in \mathcal{B}<math>
• <math>U<math> is absolutely convex and absorbent.

<math>\mathcal{B}<math> is called a locally convex basis for <math>X<math>.

Conversely for every non-empty family of subsets <math>\mathcal{B}<math> of a vector space <math>X<math> with the above properties, there exists exactly one topology <math>\tau<math> so that <math>(X,\tau)<math> is a locally convex topological vector space and <math>\mathcal{B}<math> a locally convex basis.

Examples

• Every Banach space is a locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of Banach spaces.

• More generally normed space is locally convex, since the triangle inequality ensures that all balls are convex.

• The space <math>(\omega,P)<math> of real valued sequences <math>\omega<math> with the family of seminorms <math>P<math> defined as

<math>P:=\{p_i \mid i \in \mathbb{N} \quad p_i:(x_k)_{k \in \mathbb{N}} \mapsto \vert x_i \vert \}<math>

• L^p spaces with <math>p \ge 1<math> are locally convex.

• Any vector space X (with or without an existing topology) can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals on X continuous. This can be seen as the weak topology defined by the algebraic dual of X.

Smooth functions

Spaces of differentiable functions give other non-normable examples. For instance, consider an open set U in Rⁿ and the set X = C^∞(U) of smooth functions f : U → R. We first define a collection of seminorms on X, and the topology will then be defined as the coarsest topology which refines the topology defined by each of the seminorms. For a compact set K and a multi-index m = (m₁, ..., m_n) we define the (K, m) semi-norm to be the supremum of the differentiation first by x₁ m₁ times, then by x₂ m₂ times and so on K. With this topology, a sequence (f_n) in X has limit f if and only if on every compact set all derivatives of f_n converge uniformly to the corresponding derivative of f. With all such semi-norms, the space X = C^∞(U) is a locally convex topological vector space, commonly denoted E(U).

Continuous functions

More abstractly, given a topological space X, the space C(X) of continuous (not necessarily bounded) functions on X can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms φ_K(f) = max { |f(x)| : x ∈ K } (as K varies over the directed set of all compact subsets of X). When X is locally compact (e.g. an open set in Rⁿ) the Stone-Weierstrass theorem applies -- any subalgebra of C(X) that separates points (e.g. polynomials) is dense.

Constructing a locally convex basis

A family of seminorms <math>P<math> can be used to define a locally convex basis <math>\mathcal{B}<math> for a vector space over a field <math>\mathbb{F}<math> and thus a unique locally convex topology.

Each semi norm <math>p<math> can be regarded as the gauge of an absolutely convex and absorbing set <math>A^p<math> defined as

<math>A^p := \{ x \in X : p(x) \le 1\}<math>

First we construct for each semi norm <math>p<math> the set of all p-balls as

<math>\mathcal{A}^p := \{ \alpha A^p : \alpha \in \mathbb{F}\}<math>

then we define the basis as the collection of all finite intersections between those p-balls

<math>\mathcal{B} := \{ \cap_{i=1}^n A_i^p : A_i^p \in \mathcal{A}^p \}<math>

All sets in <math>\mathcal{B}<math> are absolutely convex and absorbing and satisfy the other properties for a locally convex basis by construction.

Continuous linear mappings

The continuous linear mappings between two topological vector spaces can be characterized using the family of semi norms.

Given two locally convex spaces <math>(X, P)<math> and <math>(Y, Q)<math> and a linear mapping <math>T<math> between them, the following conditions are equivalent

• <math>T<math> is continuous
• for every semi norm <math>q<math> in <math>Q<math>, <math>q \circ T<math> is continuous
• for every semi norm <math>q<math> in <math>Q<math> we can construct a new semi norm <math>p'<math> in <math>(X,P)<math> so that

<math>q(Tx) \le p'(x) \qquad (x \in X)<math>

with

<math>p'(x) := M \sum_{i=1}^n p_i(x)<math>

for a positive real number <math>M<math> and <math>n<math> semi norms in <math>P<math>

Properties

Given a vector space X a family P of seminorms is called total if

<math>\forall x \in X \setminus \{0\} \quad \exists p \in P : p(x) \neq 0<math>

The topology for a locally convex space is Hausdorff if and only if the family of seminorms is total.

A locally convex space is seminormable if and only if there exists a bounded neighbourhood for zero.

A locally convex space is semimetrizeable if and only if the topology can be defined by a countable family of semi-norms.de:Lokal_konvexer_Raum