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. A topological homomorphism or simply homomorphism (if no confusion will arise) is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when which is the of is given the subspace topology induced by 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 is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. Suppose that is a linear map between TVSs and note that can be decomposed into the composition of the following canonical linear maps: where is the canonical quotient map and is the inclusion map. The following are equivalent: is a topological homomorphism for every neighborhood base of the origin in is a neighborhood base of the origin in the induced map is an isomorphism of TVSs If in addition the range of is a finite-dimensional Hausdorff space then the following are equivalent: is a topological homomorphism is continuous is continuous at the origin is closed in 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. Every continuous linear functional on a TVS is a topological homomorphism. Let be a -dimensional TVS over the field and let be non-zero. Let be defined by If has it usual Euclidean topology and if is Hausdorff then is a TVS-isomorphism.