Hilbert Space

**charikaar** · Oct 25th 2009, 06:07 AM

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
_________________

**Jose27** · Oct 25th 2009, 09:18 AM

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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