Topological homomorphism


In functional analysis, a topological homomorphism or simply homomorphism is a continuous linear map u : X → Y between topological vector spaces such that the induced map u : X → Im u is an open mapping when Im u, which is the range or 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 homorphism.

Characterizations

Suppose that u : X → Y is a linear map between TVSs and note that u can be decomposed into the composition of the following canonical linear maps:
where is the canonical quotient map and is the natural inclusion.
The following are equivalent:

  1. u is a topological homomorphism;
  2. for every neighborhood base of the origin in X, is a neighborhood base of the origin in Y;
  3. the induced map 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. is closed in X.

Sufficient conditions

Theorem:
Let u : X → Y be a continuous linear map from an LF-space X into a TVS Y.
If Y is also an LF-space or if Y is a Fréchet space then u : X → Y is a topological homomorphism.

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 u : X → Y be a continuous linear map between two complete metrizable TVSs.
If is a dense subset of Y then either is meager in Y or else u : X → Y is a surjective topological homomorphism.
In particular, u : X → Y is a topological homomorphism if and only if is a closed subset of Y.
Corollary:
Let and be two TVS topologies on a vector space X such that both and are complete metrizable TVSs. If is finer than then.
Corollary:
If X is a complete metrizable TVS, M and N are two closed vector subspaces of X, and if X is the algebraic direct sum of M and N, then X is the direct sum of M and N in the category of topological vector spaces.

Examples