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Math Help - Hilbert Space

  1. #1
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    Hilbert Space

    Can anyone give me some help on this one please.

    Let H be a Hilbert space and Let be an orthonormal sequence for H. Write down Parseval's identity for . Show that an orthonormal basis in H if and only if it satisfies Parseval's identity.

    Parseval's identity is how do i write using this identity?

    Thanks
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  2. #2
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    It's easy to see that the sequence satisfies Parseval's identity iff \sum_{k=1} ^{\infty} <x,e_k>e_k = x for all x \in H. To prove that this happens iff the sequence is an orthonormal basis see here (the last post).
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