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Math Help - Hilbert Space

  1. #1
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    Hilbert Space

    Let H be a Hilbert space. Show that H^* is also a Hilbert space with respect to the inner product defined by (f_x,f_y) = (y,x).
    ( x,y \in H  , f_y,f_x \in H^* )

    I had proven that it is an inner product space and my problem is how to show that H^* is a Hilbert space.

    Can anyone help?
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  2. #2
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    Frechet-Riesz theorem assures that H^* is a Banach space with the operator norm.

    Another thing you could use is that if X,Y are normed spaces and Y is a Banach space then \mathcal{B} (X,Y) := \{ f:X\rightarrow Y : f \ is \ linear \ and \ continous \} is also a Banach space with the operator norm (which in your case coincides with the one induced by the inner product).
    Last edited by Jose27; November 10th 2009 at 10:37 AM.
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