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 .

literature

• Dirk Werner : Functional Analysis . 6th corrected edition. Springer, Berlin 2007, ISBN 978-3-540-72533-6 .