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Riesz' lemma gives us that in infinite-dimensional spaces no ball is compact. but what is 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?

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Riesz's Lemma says given a proper closed linear subspace $Y$ of the normed linear space $X$ and $0<\theta<1$, there is an element $x$ of norm $1$ so that $\Vert x-y\Vert>\theta$ for all $y\in Y$. One then can construct a sequence in the unit sphere of $X$ that is separated. –  David Mitra Jan 2 at 14:31

3 Answers 3

up vote 8 down vote accepted

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.

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particularly nice proof, thanks. –  Xin Wang Jan 2 at 15:23

Hint: In a metric space, sequential compactness and compactness are equivalent. Now consider a sequence consisting of unit length basis vectors.

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Sounds right to me. The sequence of points $(1,0,0,\ldots), (0, 1, 0, \ldots), (0, 0, 1, \ldots), \ldots$ is a sequence of points on the sphere that has no convergent subsequence, because the distance between any two of the points is $\sqrt{2}$.

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