Topological homomorphism

In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map u : X → Y between topological vector spaces (TVSs) 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.[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 homorphism.

Definitions

Definition:[2] 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 → Y is a linear map between TVSs and note that u can be decomposed into the composition of the following canonical linear maps:

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

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 natural inclusion. The following are equivalent:

1. u is a topological homomorphism;
2. for every neighborhood base ${\displaystyle {\mathcal {U}}}$ of the origin in X, ${\displaystyle u\left({\mathcal {U}}\right)}$ is a neighborhood base of the origin in 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 u is a finite-dimensional Hausdorff space then the following are equivalent:

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

Sufficient conditions

Theorem:[1] 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:[1] Let u : X → Y be a continuous linear map between two complete metrizable TVSs. If ${\displaystyle \operatorname {Im} u}$ is a dense subset of Y then either ${\displaystyle \operatorname {Im} u}$ is meager (i.e. of the first category) 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 ${\displaystyle \operatorname {Im} u}$ is a closed subset of Y.

Corollary:[1] Let ${\displaystyle \tau _{1}}$ and ${\displaystyle \tau _{2}}$ be two TVS topologies on a vector space X such that both ${\displaystyle \left(X,\tau _{1}\right)}$ and ${\displaystyle \left(X,\tau _{2}\right)}$ are complete metrizable TVSs. If ${\displaystyle \tau _{2}}$ is finer than ${\displaystyle \tau _{1}}$ (or if ${\displaystyle \tau _{1}}$ is finer than ${\displaystyle \tau _{2}}$) then ${\displaystyle \tau _{1}=\tau _{2}}$.

Corollary:[1] 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 (i.e. the direct sum in the category of vector spaces), then X is the direct sum of M and N in the category of topological vector spaces.

Examples

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

See also

• Homomorphism – Structure-preserving map between two algebraic structures of the same type
• Open mapping

References

1. Schaefer 1999, pp. 74–78. sfn error: no target: CITEREFSchaefer1999 (help)
2. Köthe 1969, p. 91. sfn error: no target: CITEREFKöthe1969 (help)

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• Köthe, Gottfried (December 19, 1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.CS1 maint: ref=harv (link) CS1 maint: date and year (link)
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