Topological homomorphism
http://dbpedia.org/resource/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.
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Topological homomorphism
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Theorem
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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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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
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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.
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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.
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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.
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Let and be TVS topologies on a vector space such that each topology makes into a complete metrizable TVSs. If either or then
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