Parseval's identity

In functional analysis, Parseval's identity, also known as Parseval's equality, is the Pythagorean theorem for inner-product spaces. It states that if B is an orthonormal basis in a complete inner-product space (i.e. a Hilbert space), then

<math>\|x\|^2=\langle x,x\rangle=\sum_{v\in B}\left|\langle x,v\rangle\right|^2.</math>

The origin of the name is in Parseval's theorem for Fourier series, which is a special case.

Parseval's identity can be proved using the Riesz-Fischer theorem.

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