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Is it true?

  1. Mar 9, 2006 #1

    dextercioby

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    Is the fact that a unit ball in an infinite-dimensional (Banach) space is a noncompact topological space...?

    If it is, how would one go about proving it...?

    Daniel.
     
  2. jcsd
  3. Mar 9, 2006 #2

    George Jones

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    I found a proof in Kreyszig.

    Every Banach space is a metric space, and in a compact subset of a metric space, every sequence has a convergent subsequence.

    Kreyszig assumes that the closed unit ball of an infinite-dimensional Banach space is compact. He then uses Riesz's Lemma (which isn't the Riesz Representation Theorem) to construct a sequence that doesn't have convergent subsequence.

    Conclusion: the closed unit ball of a Banach space is compact if and only if the Banach space is finite-dimensional.

    Regards,
    George
     
  4. Mar 10, 2006 #3

    dextercioby

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    Thank you, George.

    Daniel.
     
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