# Compactness and Normed Linear spaces

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?

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.