Are any two isomorphic normed linear spaces homeomorphic?

Question

We know that any two finite-dimensional linear spaces (over a same field) of same dimension is isomorphic.

Qn.1 Are any two finite-dimensional normed linear spaces (over a same field) with same dimension isometrically isomorphic?

Qn.2 Are any two isomorphic normed linear spaces are homeomorphic?

Thanks in advance. Any answer would be appreciated.

Answer to Qn 1. $\mathbb R^{2}$ with Euclidean norm is not isometrically isomorphic to $\mathbb R^{2}$ with the max norm. — Kavi Rama Murthy, Apr 11 '19 at 10:25

score 3 · Answer 1 · answered Apr 11 '19 at 10:27

3

No. The two dimensional normed space whose "unit ball" is a square is not isometric to the Euclidean two dimensional space.
Any two finite dimensional normed linear spaces (of the same dimension) are homeomorphic, because any two convex,compact sets in $\mathbb{R}^n$ with non-empty interior are homeomorphic one to another.

The proofs are not too difficult.

For infinite dimension there are positive results, too. See here.

answered Apr 11 '19 at 10:27

uniquesolution

Can you provide proof (link/something) on the claim "Any two finite dimensional normed linear spaces (of the same dimension) are homeomorphic?" – xyz Nov 10 '22 at 22:23

1 Answers1