About: Topological homomorphism

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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.

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rdfs:label	Topological homomorphism (en)
rdfs:comment	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. (en)
name	Theorem (en) Corollary (en)
dcterms:subject	Functional analysis
Wikipage page ID	63987005 (xsd:integer)
Wikipage revision ID	1097566098 (xsd:integer)
Link from a Wikipage to another Wikipage	Monomorphism Topological embedding Homomorphism Inclusion map Complete topological vector space Continuous linear functional Continuous map Meager set Fréchet space Stefan Banach Functional analysis Meagre set Topological vector space Hausdorff space Linear map F-space Functional analysis LF-space Metrizable topological vector space Open mapping Open mapping theorem (functional analysis) Euclidean topology Image (mathematics) Subspace topology Quotient map Injective map
sameAs	Topological homomorphism Topological homomorphism
dbp:wikiPageUsesTemplate	dbt:Valdivia_Topics_in_Locally_Convex_Spaces dbt:Anchor dbt:Annotated_link dbt:Reflist dbt:Sfn dbt:Bourbaki_Topological_Vector_Spaces_Part_1_Chapters_1–5 dbt:Jarchow_Locally_Convex_Spaces dbt:Köthe_Topological_Vector_Spaces_I dbt:Köthe_Topological_Vector_Spaces_II dbt:Math_theorem dbt:Narici_Beckenstein_Topological_Vector_Spaces dbt:Robertson_Topological_Vector_Spaces dbt:Schaefer_Wolff_Topological_Vector_Spaces dbt:Schechter_Handbook_of_Analysis_and_Its_Foundations dbt:Swartz_An_Introduction_to_Functional_Analysis dbt:TopologicalVectorSpaces dbt:Trèves_François_Topological_vector_spaces,_distributions_and_kernels dbt:Wilansky_Modern_Methods_in_Topological_Vector_Spaces dbt:Voigt_A_Course_on_Topological_Vector_Spaces
has abstract	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. (en)
math statement	Let be a continuous linear map between two complete metrizable TVSs. If which is the range of is a dense subset of then either is meager in or else is a surjective topological homomorphism. In particular, is a topological homomorphism if and only if is a closed subset of (en) If is a complete metrizable TVS, and are two closed vector subspaces of and if is the algebraic direct sum of and , then is the direct sum of and in the category of topological vector spaces. (en) Suppose be a continuous linear operator between two Hausdorff TVSs. If is a dense vector subspace of and if the restriction to is a topological homomorphism then is also a topological homomorphism. So if and are Hausdorff completions of and respectively, and if is a topological homomorphism, then 's unique continuous linear extension is a topological homomorphism. (en) Let be a surjective continuous linear map from an LF-space into a TVS If is also an LF-space or if is a Fréchet space then is a topological homomorphism. (en) Let and be TVS topologies on a vector space such that each topology makes into a complete metrizable TVSs. If either or then (en)
prov:wasDerivedFrom	wikipedia-en:Topological_homomorphism?oldid=1097566098&ns=0
page length (characters) of wiki page	8324 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Topological_homomorphism
is Link from a Wikipage to another Wikipage of	Topological vector space homomorphism Riesz space Inductive tensor product Complete topological vector space Nuclear operator Topological vector space F. Riesz's theorem Injective tensor product Open mapping theorem (functional analysis) Order topology (functional analysis) TVS-homomorphism TVS homomorphism
is Wikipage redirect of	Topological vector space homomorphism TVS-homomorphism TVS homomorphism
is foaf:primaryTopic of	wikipedia-en:Topological_homomorphism

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