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 $$u : X \to Y$$ between topological vector spaces (TVSs) such that the induced map $$u : X \to \operatorname{Im} u$$ is an open mapping when $$\operatorname{Im} u := u(X),$$ which is the image of $$u,$$ is given the subspace topology induced by $$Y.$$ 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 $$u : X \to Y$$ is a linear map between TVSs and note that $$u$$ can be decomposed into the composition of the following canonical linear maps:


 * $$X ~\overset{\pi}{\rightarrow}~ X / \operatorname{ker} u ~\overset{u_0}{\rightarrow}~ \operatorname{Im} u ~\overset{\operatorname{In}}{\rightarrow}~ Y$$

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

The following are equivalent:
 * 1) $$u$$ is a topological homomorphism
 * 2) for every neighborhood base $$\mathcal{U}$$ of the origin in $$X,$$ $$u\left( \mathcal{U} \right)$$ is a neighborhood base of the origin in $$Y$$
 * 3) the induced map $$u_0 : X / \operatorname{ker} u \to \operatorname{Im} u$$ is an isomorphism of TVSs

If in addition the range of $$u$$ is a finite-dimensional Hausdorff space then the following are equivalent:
 * 1) $$u$$ is a topological homomorphism
 * 2) $$u$$ is continuous
 * 3) $$u$$ is continuous at the origin
 * 4) $$u^{-1}(0)$$ is closed in $$X$$

Sufficient conditions
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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.

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

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