F. Riesz's theorem

Statement

Consequences

• 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 M is a closed vector subspace of a TVS Y and if F is a finite-dimensional vector subspace of Y then M + F is a closed vector subspace of Y.
• Every vector space isomorphism between two finite-dimensional Hausdorff TVSs is a TVS-isomorphism.
• Uniqueness of topology: If X is a finite-dimensional vector space and if ?₁ and ?₂ are two Hausdorff TVS topologies on X then ?₁ = ?₂.
• Finite-dimensional domain: A linear map L : F → Y 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 L : X → Y with a Hausdorff finite-dimensional range is an open map and thus a topological homomorphism.
  In particular, the range of L is TVS-isomorphic to X/L⁻¹.
• A TVS X is locally compact if and only if X/ 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 closed convex hull of that set.
• A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.