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Sequence in Hilbert space,example

  1. Jul 28, 2012 #1
    Hi. I`m working on some exercises but I could`t find any clue for this one:

    Find a bounded sequence (as like the norm) in l^2 Hilbert space,that weakly converges to 0 (as like the weak topology)
    but doesn`t have any convergent subsequences (as in strong topology).
    Could someone help me?
     
  2. jcsd
  3. Aug 7, 2012 #2

    Bacle2

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    I think the unit cube in Hilbert space l^2:{ei:=δij}, i.e., ei

    has a 1 in the i-th position and is 0 otherwise, is an example: for one thing, it does

    not have a finite ε-net for ε <1/2, say, since the l2-distance is √2, so that it

    cannot have convergent subsequences -- the √2 distance is a barrier to being Cauchy.
     
  4. Aug 8, 2012 #3

    Bacle2

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    I forgot the second part, about the sequence converging weakly:

    first , in l2:

    We need to show:

    Limn→∞ <en, an>=<0,an>=0

    But notice that the product on the left equals the n-th term of a sequence an

    in l2. Since an is square-summable, it is Cauchy, so that

    its n-th term goes to zero as n→ ∞.

    But this sequence {en}=(δij)j=1,...,∞ converges weakly

    to zero in _any_ Hilbert space:

    Let h be an element in any Hilbert space H. Then, since {en} is a maximal

    orthonormal set, it is a Hamel basis for H . Then h is a finite linear combination of the

    basis elements in {en}. Let ek be the largest index in the

    linear combination. Then, when n>k , < en,h>=0.

    h=Ʃ
     
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