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