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I am supposed to prove the following result in my worksheet:

Exercise. Show that any two finite-dimensional normed spaces are homeomorphic. In particular, show that every normed space (over $\Bbb {K}$) of dimension $n$ is isomorph to $\Bbb K^n.$

My concern. One should know that, for example, $\Bbb R^n$ and $\Bbb R^m$ are not homeomorphic (see here, for $m \neq n$). Since $\Bbb R^n$ and $\Bbb R^m$ are normed spaces (there are several known norms defined on such spaces) with finite dimension ($n$ and $m$, respectively) this contradicts the statement (?).

I believe that for finite-dimensional normed spaces with the same dimension the result is valid and it isn't hard to prove. But like this, I can't see how this is true.

Thanks for any help in advance.

xyz
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    Indeed the statement isn't correct. The spaces must have the same dimension to be homeomorphic. – Falcon Nov 06 '22 at 01:07
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    The second half of the exercise makes it sound like the original intent was indeed to show that any two finite-dimensional spaces of the same dimension $n$ are homeomorphic (and thus in particular are homeomorphic to $k^n$ under the usual norm). – anomaly Nov 06 '22 at 01:11
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    Yup... Probably just a typo! Thanks for the feedback. – xyz Nov 06 '22 at 01:36

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