Parseval's theorem

Parseval's theorem says that the norm of a function can be recovered by its Fourier coefficients (L.6.4.1 ):

$$\frac{1}{2\pi} = \int_{-\pi}^\pi |f(x)|^2 dx = \sum_{k\in \mathbb Z} |c_k|^2$$

where

$$c_k:= \frac{1}{2\pi}\int_{-\pi}^\pi f(x) e ^{ikx}dx$$.

But in fact Parseval's theorem is a more general theorem -- it says that, given any orthonormal basis for a space $$V$$, the norm of an element in $$V$$ can be recovered by its coefficients wrt this basis (see (iv) in Th. 4.7.2 ).