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 closed convex hull of that set.
- A Hausdorff locally bounded TVS with the Heine-Borel property is necessarily finite-dimensional.
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