F. Riesz's theorem

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F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

Contents

Statement

Recall that a topological vector space (TVS) is Hausdorff if and only if the singleton set consisting entirely of the origin is a closed subset of A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a linear homeomorphism.

F. Riesz theorem [1] [2]   A Hausdorff TVS over the field ( is either the real or complex numbers) is finite-dimensional if and only if it is locally compact (or equivalently, if and only if there exists a compact neighborhood of the origin). In this case, is TVS-isomorphic to

Consequences

Throughout, are TVSs (not necessarily Hausdorff) with a finite-dimensional vector space.

In particular, the range of is TVS-isomorphic to

See also

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