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.

  • 1
    $\begingroup$ Answer to Qn 1. $\mathbb R^{2}$ with Euclidean norm is not isometrically isomorphic to $\mathbb R^{2}$ with the max norm. $\endgroup$ – Kavi Rama Murthy Apr 11 at 10:25
  1. No. The two dimensional normed space whose "unit ball" is a square is not isometric to the Euclidean two dimensional space.

  2. 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.


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