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 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

• Bessel's inequality

References

• Johnson & Riess, Numerical Analysis. ISBN 0–201–10392–3.