Bessel's inequality

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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 .


If a Hilbert space and an orthonormal system , then the inequality holds for all

where represents the scalar product on the Hilbert space.

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 .