# Bessel's inequality

In functional analysis, Bessel's inequality describes the fact that a vector of a Hilbert space is at least as “long” as its orthogonal projection onto any sub-vector space . It is named after the German mathematician Friedrich Wilhelm Bessel , who proved it in 1828 for the special case of the Fourier series .

## statement

If a Hilbert space and an orthonormal system , then the inequality holds for all${\ displaystyle H}$${\ displaystyle S \ subset H}$${\ displaystyle x \ in H}$

${\ displaystyle \ sum _ {e \ in S} \ vert \ langle x, e \ rangle \ vert ^ {2} \ leq \ Vert x \ Vert ^ {2},}$

where represents the scalar product on the Hilbert space. ${\ displaystyle \ langle \ cdot, \ cdot \ rangle}$

If the orthonormal system is even an orthonormal basis , then equality always applies. The relation is then called Parseval's equation and represents a generalization of the Pythagorean theorem for Prähilbert dreams .