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
\|x\|^2=\langle x,x\rangle=\sum_{v\in B}\langle x,v\rangle^2.
The origin of the name is in
Parseval's theorem
for
Fourier series
, which is a special case.
See also
Bessel's inequality
References
Johnson & Riess,
Numerical Analysis
. ISBN 0-201-10392-3.
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