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 between topological vector spaces (TVSs) such that the induced map is an open mapping when which is the image of is given the subspace topology induced by [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 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:

  1. is a topological homomorphism;
  2. for every neighborhood base of the origin in is a neighborhood base of the origin in ;[1]
  3. the induced map is an isomorphism of TVSs.[1]

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

  1. is a topological homomorphism;
  2. is continuous;[1]
  3. is continuous at the origin;[1]
  4. is closed in [1]

Sufficient conditions

Theorem[1] — 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.

Theorem[3] — 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.[3]

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. (However, it is possible for to be surjective but for 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 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 (that is, of the first category) in or else is a surjective topological homomorphism. In particular, is a topological homomorphism if and only if is a closed subset of

Corollary[4] — Let and be TVS topologies on a vector space such that each topology makes into a complete metrizable TVSs. If either or then

Corollary[4] — If is a complete metrizable TVS, and are two closed vector subspaces of and if is the algebraic direct sum of and (i.e. the direct sum in the category of vector spaces), then is the direct sum of and in the category of topological vector spaces.

Examples

Every continuous linear functional on a TVS is a topological homomorphism.[1]

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.

See also

References

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

Bibliography