Topological homomorphism

In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

Definitions

A topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map [math]\displaystyle{ u : X \to Y }[/math] between topological vector spaces (TVSs) such that the induced map [math]\displaystyle{ u : X \to \operatorname{Im} u }[/math] is an open mapping when [math]\displaystyle{ \operatorname{Im} u := u(X), }[/math] which is the image of [math]\displaystyle{ u, }[/math] is given the subspace topology induced by [math]\displaystyle{ Y. }[/math] This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.

Characterizations

Suppose that [math]\displaystyle{ u : X \to Y }[/math] is a linear map between TVSs and note that [math]\displaystyle{ u }[/math] can be decomposed into the composition of the following canonical linear maps:

[math]\displaystyle{ X ~\overset{\pi}{\rightarrow}~ X / \operatorname{ker} u ~\overset{u_0}{\rightarrow}~ \operatorname{Im} u ~\overset{\operatorname{In}}{\rightarrow}~ Y }[/math]

where [math]\displaystyle{ \pi : X \to X / \operatorname{ker} u }[/math] is the canonical quotient map and [math]\displaystyle{ \operatorname{In} : \operatorname{Im} u \to Y }[/math] is the inclusion map.

The following are equivalent:

1. [math]\displaystyle{ u }[/math] is a topological homomorphism
2. for every neighborhood base [math]\displaystyle{ \mathcal{U} }[/math] of the origin in [math]\displaystyle{ X, }[/math] [math]\displaystyle{ u\left( \mathcal{U} \right) }[/math] is a neighborhood base of the origin in [math]\displaystyle{ Y }[/math]
3. the induced map [math]\displaystyle{ u_0 : X / \operatorname{ker} u \to \operatorname{Im} u }[/math] is an isomorphism of TVSs

If in addition the range of [math]\displaystyle{ u }[/math] is a finite-dimensional Hausdorff space then the following are equivalent:

1. [math]\displaystyle{ u }[/math] is a topological homomorphism
2. [math]\displaystyle{ u }[/math] is continuous
3. [math]\displaystyle{ u }[/math] is continuous at the origin
4. [math]\displaystyle{ u^{-1}(0) }[/math] is closed in [math]\displaystyle{ X }[/math]

Sufficient conditions

Theorem — Let [math]\displaystyle{ u : X \to Y }[/math] be a surjective continuous linear map from an LF-space [math]\displaystyle{ X }[/math] into a TVS [math]\displaystyle{ Y. }[/math] If [math]\displaystyle{ Y }[/math] is also an LF-space or if [math]\displaystyle{ Y }[/math] is a Fréchet space then [math]\displaystyle{ u : X \to Y }[/math] is a topological homomorphism.

Theorem — Suppose [math]\displaystyle{ f : X \to Y }[/math] be a continuous linear operator between two Hausdorff TVSs. If [math]\displaystyle{ M }[/math] is a dense vector subspace of [math]\displaystyle{ X }[/math] and if the restriction [math]\displaystyle{ f\big\vert_M : M \to Y }[/math] to [math]\displaystyle{ M }[/math] is a topological homomorphism then [math]\displaystyle{ f : X \to Y }[/math] is also a topological homomorphism.

So if [math]\displaystyle{ C }[/math] and [math]\displaystyle{ D }[/math] are Hausdorff completions of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y, }[/math] respectively, and if [math]\displaystyle{ f : X \to Y }[/math] is a topological homomorphism, then [math]\displaystyle{ f }[/math]'s unique continuous linear extension [math]\displaystyle{ F : C \to D }[/math] is a topological homomorphism. (However, it is possible for [math]\displaystyle{ f : X \to Y }[/math] to be surjective but for [math]\displaystyle{ F : C \to D }[/math] to not be injective.)

Open mapping theorem

The open mapping theorem, also known as Banach's homomorphism theorem, gives a sufficient condition for a continuous linear operator between complete metrizable TVSs to be a topological homomorphism.

Theorem — Let [math]\displaystyle{ u : X \to Y }[/math] be a continuous linear map between two complete metrizable TVSs. If [math]\displaystyle{ \operatorname{Im} u, }[/math] which is the range of [math]\displaystyle{ u, }[/math] is a dense subset of [math]\displaystyle{ Y }[/math] then either [math]\displaystyle{ \operatorname{Im} u }[/math] is meager (that is, of the first category) in [math]\displaystyle{ Y }[/math] or else [math]\displaystyle{ u : X \to Y }[/math] is a surjective topological homomorphism. In particular, [math]\displaystyle{ u : X \to Y }[/math] is a topological homomorphism if and only if [math]\displaystyle{ \operatorname{Im} u }[/math] is a closed subset of [math]\displaystyle{ Y. }[/math]

Corollary — Let [math]\displaystyle{ \sigma }[/math] and [math]\displaystyle{ \tau }[/math] be TVS topologies on a vector space [math]\displaystyle{ X }[/math] such that each topology makes [math]\displaystyle{ X }[/math] into a complete metrizable TVSs. If either [math]\displaystyle{ \sigma \subseteq \tau }[/math] or [math]\displaystyle{ \tau \subseteq \sigma }[/math] then [math]\displaystyle{ \sigma = \tau. }[/math]

Corollary — If [math]\displaystyle{ X }[/math] is a complete metrizable TVS, [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] are two closed vector subspaces of [math]\displaystyle{ X, }[/math] and if [math]\displaystyle{ X }[/math] is the algebraic direct sum of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] (i.e. the direct sum in the category of vector spaces), then [math]\displaystyle{ X }[/math] is the direct sum of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] in the category of topological vector spaces.

Examples

Every continuous linear functional on a TVS is a topological homomorphism.

Let [math]\displaystyle{ X }[/math] be a [math]\displaystyle{ 1 }[/math]-dimensional TVS over the field [math]\displaystyle{ \mathbb{K} }[/math] and let [math]\displaystyle{ x \in X }[/math] be non-zero. Let [math]\displaystyle{ L : \mathbb{K} \to X }[/math] be defined by [math]\displaystyle{ L(s) := s x. }[/math] If [math]\displaystyle{ \mathbb{K} }[/math] has it usual Euclidean topology and if [math]\displaystyle{ X }[/math] is Hausdorff then [math]\displaystyle{ L : \mathbb{K} \to X }[/math] is a TVS-isomorphism.

See also

• Homomorphism – Structure-preserving map between two algebraic structures of the same type
• Surjection of Fréchet spaces – Characterization of surjectivity
• Topological vector space – Vector space with a notion of nearness

Bibliography

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