Parseval's identity
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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 an inner-product space, then
- <math>\|x\|^2=\langle x,x\rangle=\sum_{v\in B}\langle x,v\rangle^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.
See also
References
- Johnson & Riess, Numerical Analysis. ISBN 0-201-10392-3.