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 $u:X\to \mathrm{Im}u$ is an open mapping when $\mathrm{Im}u:=u(X),$ which is the image of $u,$ is given the subspace topology induced by $Y.$Template:Sfn 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 embeddingTemplate:Anchor or a topological monomorphismTemplate:Sfn 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:

$X\stackrel{\pi }{\to }X/\ker u\stackrel{u_0}{\to }\mathrm{Im}u\stackrel{\mathrm{In}}{\to }Y$

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

The following are equivalent:

1. $u$ is a topological homomorphism
2. for every neighborhood base $𝒰$ of the origin in $X,$ $u\left(𝒰\right)$ is a neighborhood base of the origin in $Y$Template:Sfn
3. the induced map $u_0:X/\ker u\to \mathrm{Im}u$ is an isomorphism of TVSsTemplate:Sfn

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 continuousTemplate:Sfn
3. $u$ is continuous at the originTemplate:Sfn
4. $u^{-1}(0)$ is closed in $X$Template:Sfn

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.Template:Sfn

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

See also

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Bibliography

• Template:Bourbaki Topological Vector Spaces Part 1 Chapters 1–5
• Template:Jarchow Locally Convex Spaces
• Template:Köthe Topological Vector Spaces I
• Template:Köthe Topological Vector Spaces II
• Template:Narici Beckenstein Topological Vector Spaces
• Template:Robertson Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Schechter Handbook of Analysis and Its Foundations
• Template:Swartz An Introduction to Functional Analysis
• Template:Swartz An Introduction to Functional Analysis
• Template:Trèves François Topological vector spaces, distributions and kernels
• Template:Valdivia Topics in Locally Convex Spaces
• Template:Voigt A Course on Topological Vector Spaces
• Template:Wilansky Modern Methods in Topological Vector Spaces