Parseval's identity

Think of a signal's total energy as the integral of its square. Parseval's identity says energy is conserved when you switch from the time picture to the frequency picture: the energy you compute by integrating the function equals the energy you compute by summing the squared sizes of its Fourier coefficients. Nothing is lost in translation between a function and its spectrum.

In the standard normalization, for f in L2 of [-pi, pi] with complex coefficients c_n, (1/(2*pi)) integral over [-pi, pi] of |f(x)|^2 dx = sum over all n of |c_n|^2. In real form it reads (1/pi) integral of |f|^2 = a_0^2/2 + sum of (a_n^2 + b_n^2). Geometrically it is the Pythagorean theorem in infinite dimensions: the squared length of a vector equals the sum of the squared lengths of its components along an orthonormal basis.

Crucially, the equality (rather than mere inequality) holds because the trigonometric system is complete in L2 — it leaves no ‘direction’ unaccounted for. If you used only part of an orthonormal system, you would get Bessel's inequality (a one-sided bound) instead. So Parseval is precisely the statement that the Fourier basis captures all the energy of every square-integrable function.