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This was left as an unproved theorem in our class:

Theroem: If $X$ is a finite dimensional normed vector space then each subset $M$ of $X$ is compact if and only if $M$ is closed and bounded.

How do I prove it? I know that $M$ is also finte dimensional and hence complete. So it is bounded.

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    If $X$ has dimension $n$, then $X$ is isomorphic to Euclidean $n$-space. Use the Heine-Borel Theorem.2012-04-10
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    In your theorem, did you want to say: "If X is a finite dimensional normed vector space then *a* subset $M$ of $X$ is compact if and only if $M$ is closed and bounded."?2012-04-10
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    @DavidMitra: Would the proof then be a one line proof?2012-04-10
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    Yes, more or less, as long as you've already proven the two ingredients (Heine-Borel for $\Bbb R^n$ and the fact that two finite dimensional normed spaces of the same dimension are isomorphic).2012-04-10
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    Yes.${}{}{}{}{}{}$2012-04-10
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    Ok. I've only seen Heine Borel for $\mathbb{R}$. The other fact has already been proven in class.2012-04-10

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