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If the set $S=\{ x \in X : ||x||=1 \}$ in the normed linear space $X$ is compact, how can it be shown that $X$ is finite dimensional?

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Following up on Martin's comment, if you know Riesz's lemma, you can use it, supposing that $X$ is infinite-dimensional, to inductively create a sequence $\{x_n\}$ of unit vectors with the property that $x_n$ has distance at least $\tfrac{1}{2}$ to $\mathrm{span} \{x_1,\dots,x_{n-1}\}$ and thus show that no subsequence of $\{x_n\}$ is Cauchy. See this blog post for the full argument.

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    $\begingroup$ The blog post assumes that the unit ball is compact, not the unit sphere, but the argument is exactly the same. $\endgroup$ – Martin Jun 19 '13 at 13:25
  • $\begingroup$ Good point, thanks. $\endgroup$ – fuglede Jun 19 '13 at 13:36

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