In mathematics, 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.
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.
Consequences
Throughout, are TVSs (not necessarily Hausdorff) with a finite-dimensional vector space.
- Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace.
- All finite-dimensional Hausdorff TVSs are Banach spaces and all norms on such a space are equivalent.
- Closed + finite-dimensional is closed: If is a closed vector subspace of a TVS and if is a finite-dimensional vector subspace of ( and are not necessarily Hausdorff) then is a closed vector subspace of
- Every vector space isomorphism (i.e. a linear bijection) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism.
- Uniqueness of topology: If is a finite-dimensional vector space and if and are two Hausdorff TVS topologies on then
- Finite-dimensional domain: A linear map between Hausdorff TVSs is necessarily continuous.
- In particular, every linear functional of a finite-dimensional Hausdorff TVS is continuous.
- Finite-dimensional range: Any continuous surjective linear map with a Hausdorff finite-dimensional range is an open map and thus a topological homomorphism.
In particular, the range of is TVS-isomorphic to
- A TVS (not necessarily Hausdorff) is locally compact if and only if is finite dimensional.
- The convex hull of a compact subset of a finite-dimensional Hausdorff TVS is compact.
- This implies, in particular, that the convex hull of a compact set is equal to the convex hull of that set.
- A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.
See also
References
Bibliography