Topological homomorphism


In functional analysis, a topological homomorphism or simply homomorphism is the analog of homomorphisms for the category of topological vector spaces.
This concept is of considerable importance in functional analysis and the famous Open [mapping theorem (functional analysis)|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 is a continuous linear map between topological vector spaces such that the induced map is an open mapping when which is the image of is given the subspace topology induced by
This concept is of considerable importance in functional analysis and the famous Open mapping [theorem (functional analysis)|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 is a linear map between TVSs and note that can be decomposed into the composition of the following canonical linear maps:
where is the canonical quotient map and is the inclusion map.
The following are equivalent:
  1. is a topological homomorphism
  2. for every neighborhood base of the origin in is a neighborhood base of the origin in
  3. the induced map is an isomorphism of TVSs
If in addition the range of is a finite-dimensional Hausdorff space then the following are equivalent:
  1. is a topological homomorphism
  2. is continuous
  3. is continuous at the origin
  4. is closed in

Sufficient conditions

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.

Examples

Every continuous linear functional on a TVS is a topological homomorphism.
Let be a -dimensional TVS over the field and let be non-zero. Let be defined by If has it usual Euclidean topology and if is Hausdorff then is a TVS-isomorphism.