The
identity is related to the
Pythagorean theorem in the more general setting of a
separable Hilbert space as follows. Suppose that H is a Hilbert space with
inner product \langle \,\cdot\,, \,\cdot\, \rangle. Let \left(e_n\right) be an
orthonormal basis of H; i.e., the
linear span of the e_n is
dense in H, and the e_n are mutually orthonormal: :\langle e_m, e_n\rangle = \begin{cases} 1 & \mbox{if}~ m = n \\ 0 & \mbox{if}~ m \neq n. \end{cases} Then Parseval's identity asserts that for every x \in H, \sum_n \left|\left\langle x, e_n \right\rangle\right|^2 = \|x\|^2. This is directly analogous to the
Pythagorean theorem in
Euclidean geometry, which asserts that the sum of the squares of the components of a vector in an orthonormal basis is equal to the squared length of the vector. One can recover the Fourier series version of Parseval's identity by letting H be the Hilbert space L^2[-\pi, \pi], and setting e_n = e^{i n x} for n \in \Z. More generally, Parseval's identity holds for arbitrary
Hilbert spaces, not necessarily separable. When the Hilbert space is not separable any orthonormal basis is uncountable and we need to generalize the concept of a series to an unconditional sum as follows: let \{e_r\}_{r\in \Gamma} an orthonormal basis of a Hilbert space (where \Gamma have arbitrary cardinality), then we say that \sum_{r\in \Gamma} a_r e_r converges unconditionally if for every \epsilon>0 there exists a finite subset A\subset \Gamma such that \left\| \sum_{r\in B}a_re_r-\sum_{r\in C}a_r e_r\right\| for any pair of finite subsets B,C\subset\Gamma that contains A (that is, such that A\subset B\cap C). Note that in this case we are using a
net to define the unconditional sum. == See also ==