A normed vector space has compact closed unit ball if and only if it is finite-dimensional. In an infinite-dimensional space, repeated application of Riesz lemma produces unit vectors separated pairwise by a fixed positive distance, contradicting sequential compactness.

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  1. Riesz lemma
  2. Banach space
  3. Normed vector space
  4. Functional analysis
  5. Analysis
  6. Area of mathematics
  7. Mathematics
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