Bessel's inequality

The Bessel inequality describes the functional analysis of the facts that a vector of a Hilbert space as its orthogonal projection is at least as " long" on an arbitrary subspace. It is named after the German mathematician Friedrich Wilhelm Bessel, who proved in 1828 for the special case of the Fourier series.

Statement

So is a Hilbert space and an orthonormal system, then for all the inequality

And the dot showing on the Hilbert space.

If the orthonormal system even an orthonormal basis, so even is global equality and the equation is called Parseval's equation and represents a generalization of the Pythagorean theorem for inner product spaces dar.