Riesz' lemma gives us that in infinite-dimensional spaces no ball is compact, but what about the sphere $\{x \in X : \|x\| = 1\}$? Can we say something about the compactness of the sphere in infinite-dimensional spaces?
(I guess the sphere is also not compact and I think one can also show this by constructing a sequence with Riesz lemma that has no convergent subsequence). Is this idea correct?
Best Answer
Suppose that the unit sphere $S_X$ of $(X, \| . \|_X)$ is compact. Then the unit ball of $X$ is the image of the compact set $[0,1] \times S_X$ by the continuous map $(t, v) \mapsto tv$, and hence is compact.