If is a normed space and the closed unit ball centered at zero is compact, then is finite dimensional.

**Proof**: Suppose is an infinite dimensional normed space and let be any point in with and let be the one-dimensional subspace of generated by . Recall that a finite dimensional subspace is always closed, and since is infinite dimensional, is a proper subspace of . By Riesz’s Lemma, there exists an with and for all . Let be the two-dimensional subspace generated by . There exists a such that and for all . Note that since is infinite dimensional, we can keep applying this procedure generating a sequence such that but for all . This means that our sequence can not be Cauchy, and so it can not be convergent, and so the closed unit ball of radius one can not be compact. This means that, since all points in the sequence are at least distance from one another, no subsequence can be Cauchy, so no subsequence can be convergent. Hence, if the closed unit ball of radius one is compact, then the space is finite dimensional.

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Note, also the converse implication should be true.

Comment by francescotudisco — September 19, 2011 @ 8:09 am |