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 $u:X\to \operatorname {Im} u$ is an open mapping when $\operatorname {Im} u:=u(X),$ which is the 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 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.

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

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

The following are equivalent:

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

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

$u$ is a topological homomorphism
$u$ is continuous^[1]
$u$ is continuous at the origin^[1]
$u^{-1}(0)$ is closed in $X$ ^[1]

Sufficient conditions

Theorem^[1] — Let $u:X\to Y$ be a surjective 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 $u:X\to Y$ is a topological homomorphism.

Theorem^[3] — Suppose $f:X\to Y$ be a continuous linear operator between two Hausdorff TVSs. If $M$ is a dense vector subspace of $X$ and if the restriction $f{\big \vert }_{M}:M\to Y$ to $M$ is a topological homomorphism then $f:X\to Y$ is also a topological homomorphism.^[3]

So if $C$ and $D$ are Hausdorff completions of $X$ and $Y,$ respectively, and if $f:X\to Y$ is a topological homomorphism, then $f$ 's unique continuous linear extension $F:C\to D$ is a topological homomorphism. (However, it is possible for $f:X\to Y$ to be surjective but for $F:C\to D$ to not be injective.)

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^[4] — Let $u:X\to Y$ be a continuous linear map between two complete metrizable TVSs. If $\operatorname {Im} u,$ which is the range of $u,$ is a dense subset of $Y$ then either $\operatorname {Im} u$ is meager (that is, of the first category) in $Y$ or else $u:X\to Y$ is a surjective topological homomorphism. In particular, $u:X\to Y$ is a topological homomorphism if and only if $\operatorname {Im} u$ is a closed subset of $Y.$

Corollary^[4] — Let $\sigma$ and $\tau$ be TVS topologies on a vector space $X$ such that each topology makes $X$ into a complete metrizable TVSs. If either $\sigma \subseteq \tau$ or $\tau \subseteq \sigma$ then $\sigma =\tau .$

Corollary^[4] — 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]

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

References

^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 74–78.
^ Köthe 1969, p. 91.
^ ^a ^b Schaefer & Wolff 1999, p. 116.
^ ^a ^b ^c Schaefer & Wolff 1999, p. 78.

Bibliography

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Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342.
Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704.
Köthe, Gottfried (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. Vol. 237. New York: Springer Science & Business Media. ISBN 978-0-387-90400-9. OCLC 180577972.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
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Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
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[FOOTNOTESchaeferWolff199974–78-1] ^ ^a ^b ^c ^d ^e ^f ^g ^h Schaefer & Wolff 1999, pp. 74–78.

[FOOTNOTEKöthe196991-2] Köthe 1969, p. 91.

[FOOTNOTESchaeferWolff1999116-3] Schaefer & Wolff 1999, p. 116.

[FOOTNOTESchaeferWolff199978-4] Schaefer & Wolff 1999, p. 78.

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v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category