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.

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:

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

where $\pi :X\to X/\operatorname {ker} u$ is the canonical quotient map and $\operatorname {In$ is the natural inclusion. 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]
$u^{-1}\left(0\right)$ 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 $\operatorname {Im} u$ is a dense subset of Y then either $\operatorname {Im} u$ 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 $\operatorname {Im} u$ is a closed subset of Y.

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

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]

References

Schaefer 1999, pp. 74–78.

Sources

Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.CS1 maint: ref=harv (link)
Treves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.CS1 maint: ref=harv (link)
Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158.CS1 maint: ref=harv (link)

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[FOOTNOTESchaefer199974–78-1] Schaefer 1999, pp. 74–78.

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
Main results	Closed graph theorem Hahn–Banach (hyperplane separation) Open mapping (Banach–Schauder) Uniform boundedness (Banach–Steinhaus)
Maps	Almost open Bilinear (form operator) and Sesquilinear forms Closed Compact operator Continuous and Discontinuous linear maps Densely defined Homomorphism Functionals Operator Seminorm Sublinear Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded 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 (Ultra-) Bornological Brauner (DF)-space Distinguished F-space Fréchet (tame Fréchet) Grothendieck Hilbert Infrabarreled Interpolation 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