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Math Help - convergence in Hilbert spaces

  1. #1
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    Unhappy convergence in Hilbert spaces

    I have the following problem:

    Let X be a Hilbert space with the inner product <.|.>
    Let C_n \in C(X) be a family of compact operators that converges in the norm to operator C.
    Let (x_n)_n be a sequence that converges weakly to x_0

    Does < C_n x_n | x_n> converge? If it does, where to?

    I'm completely lost, please help.
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by marianne View Post
    Let X be a Hilbert space with the inner product <.|.>
    Let C_n \in C(X) be a family of compact operators that converges in the norm to operator C.
    Let (x_n)_n be a sequence that converges weakly to x_0

    Does < C_n x_n | x_n> converge? If it does, where to?
    Three things you need to know. (1) Weakly convergent sequences are bounded. (This follows from the uniform boundedness principle, because a weakly convergent sequence is obviously weakly bounded.) (2) Compact operators convert weakly convergent sequences into norm-convergent sequences. (3) The set of compact operators is norm-closed, so that if C_n→C_0 in norm and each C_n is compact, then so is C.

    We want to show that \langle C_n x_n | x_n\rangle\to \langle C_0 x_0 | x_0\rangle. Here's an outline of how to do it. First, for n large enough, Cx_n is close to Cx_0 in norm, by (2) and (3). Next, for n large enough, C_nx_n is close to Cx_n (this follows from (1), together with the fact that C_n→C_0 in norm.

    Putting those two statements together, you see that (for n large enough) \langle C_nx_n|x_n\rangle is close to \langle C_0x_0|x_n\rangle. But the weak convergence tells you that (again for n large enough) \langle C_0x_0|x_n\rangle is close to \langle C_0x_0|x_0\rangle. Therefore \langle C_n x_n | x_n\rangle\to \langle C_0 x_0 | x_0\rangle as n→∞.
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  3. #3
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    Thank you so much!

    I hope nobody here won't mind if I post another question or two in the next couple of days. :-)
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  4. #4
    MHF Contributor kalagota's Avatar
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    THERE ARE LOTS OF PEOPLE WHO MIND THINGS..

    just kidding..
    post your questions and everyone will do his/her best to help..
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