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I know that in normed linear space closed unit ball is compact iff space is finite dimensional. (I proved this by using Riesz lemma). My question is how to prove same statement for inner product space ( without using Riesz lemma)

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    $\begingroup$ I don't know if there is another proof, but the point that makes it easier for inner product spaces is that the special case of Riesz lemma is easier to prove: you can just orthonormalize a linearly independent sequence of vectors via Gram-Schmidt. $\endgroup$ – user228113 Jun 18 '17 at 18:23
  • $\begingroup$ Every inner product space is a normed linear space. Hence, you have already proven it!? $\endgroup$ – gerw Jun 19 '17 at 10:28

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