A topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map ${\displaystyle u:X\to Y}$  between topological vector spaces (TVSs) such that the induced map ${\displaystyle u:X\to \operatorname {Im} u}$  is an open mapping when ${\displaystyle \operatorname {Im} u:=u(X),}$  which is the image of ${\displaystyle u,}$  is given the subspace topology induced by ${\displaystyle Y.}$ ^[1] 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^[2] is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.

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

${\displaystyle X~{\overset {\pi }{\rightarrow }}~X/\operatorname {ker} u~{\overset {u_{0}}{\rightarrow }}~\operatorname {Im} u~{\overset {\operatorname {In} }{\rightarrow }}~Y}$ 

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

The following are equivalent:

1. ${\displaystyle u}$  is a topological homomorphism
2. for every neighborhood base ${\displaystyle {\mathcal {U}}}$  of the origin in ${\displaystyle X,}$  ${\displaystyle u\left({\mathcal {U}}\right)}$  is a neighborhood base of the origin in ${\displaystyle Y}$ ^[1]
3. the induced map ${\displaystyle u_{0}:X/\operatorname {ker} u\to \operatorname {Im} u}$  is an isomorphism of TVSs^[1]

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

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

Open mapping theorem edit

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.

Every continuous linear functional on a TVS is a topological homomorphism.^[1]

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

• Homomorphism – Structure-preserving map between two algebraic structures of the same type
• Open mapping – A function that sends open (resp. closed) subsets to open (resp. closed) subsetsPages displaying short descriptions of redirect targets
• Surjection of Fréchet spaces – Characterization of surjectivity
• Topological vector space – Vector space with a notion of nearness

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