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.
Recall that a topological vector space (TVS)
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
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).