
Hilbert Space
Can anyone give me some help on this one please.
Let H be a Hilbert space and Let http://www.sosmath.com/CBB/latexrend...769ef52c21.gif be an orthonormal sequence for H. Write down Parseval's identity for http://www.sosmath.com/CBB/latexrend...769ef52c21.gif. Show thathttp://www.sosmath.com/CBB/latexrend...769ef52c21.gif an orthonormal basis in H if and only if it satisfies Parseval's identity.
Parseval's identity is http://www.sosmath.com/CBB/latexrend...3d5fe1a0c8.gif how do i write http://www.sosmath.com/CBB/latexrend...769ef52c21.gif using this identity?
Thanks
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It's easy to see that the sequence satisfies Parseval's identity iff $\displaystyle \sum_{k=1} ^{\infty} <x,e_k>e_k = x$ for all $\displaystyle x \in H$. To prove that this happens iff the sequence is an orthonormal basis see here (the last post).