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Math Help - Closed unit ball in B(H)

  1. #1
    Member Mauritzvdworm's Avatar
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    Closed unit ball in B(H)

    Show that in an infinite dimensional Hilbert space H the closed unit ball (B(H))_{1} in not compact in the strong operator topology
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  2. #2
    Member Focus's Avatar
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    Quote Originally Posted by Mauritzvdworm View Post
    Show that in an infinite dimensional Hilbert space H the closed unit ball (B(H))_{1} in not compact in the strong operator topology
    Pick an infinite sequence of orthonormal elements. Work out the distance between them and so conclude that it cannot have a Cauchy subsequence. Every convergent sequence is Cauchy, so this means that it cannot have a convergent subsequence.
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