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Math Help - Infinite Dimensional Banach Spaces

  1. #1
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    Infinite Dimensional Banach Spaces

    Dear Colleagues,

    I have the following question,
    Why infinite dimensional Banach spaces are not locally compact.


    Regards,

    Raed.
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  2. #2
    Super Member girdav's Avatar
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    We only have to show that the unit ball \overline B of E is not compact. If we assume that the unit ball is compact, then we can find an integer N and x_1,\ldots,x_N  \in\overline B such that \displaystyle \overline B\subset\bigcup_{j=1}^NB(x_j,\frac 12).
    If we denote by F=\mathrm{Span}(x_1,\ldots,x_N) we have \overline B \subset F+B\left(0,\frac 12\right). Now we can show by induction that \overline  B\subset F+B\left(0,\frac 1{2^n}\right). F is closed as a finite dimensional subspace hence \overline B\subset F and E=F. E is a finite dimensional space.
    Last edited by girdav; March 23rd 2011 at 11:48 AM.
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  3. #3
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    Thank you very much for your reply.
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