Parseval's identity

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In functional analysis, a branch of mathematics, Parseval's identity, also known as Parseval's equality, is the Pythagorean theorem for inner-product spaces. Let $\left(H,\langle\cdot,\cdot\rangle\right)$ be a complete inner-product space (i.e., a Hilbert space) and $B\subset H$ be an orthonormal basis in H (meaning that, for all $x\in B$ and $y\in B$ , $\langle x,y\rangle=1$ if $x = y$ and $\langle x,y\rangle=0$ if $x\ne y$ ) which is total in it (meaning that $\overline{\operatorname{span}(B)}=H))$ ). Then,

$\|x\|^2=\langle x,x\rangle=\sum_{v\in B}\left|\langle x,v\rangle\right|^2.$

The totality assumption on B is crucial for the validity of the identity. If B is not total, then the equality in Parseval's identity must be replaced by ≥, thus yielding Bessel's inequality. 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.