Inequalities | Hilbert space

Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let be a Hilbert space, and suppose that is an orthonormal sequence in . Then, for any in one has where ⟨·,·⟩ denotes the inner product in the Hilbert space . If we define the infinite sum consisting of "infinite sum" of vector resolute in direction , Bessel's inequality tells us that this series converges. One can think of it that there exists that can be described in terms of potential basis . For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently with ). Bessel's inequality follows from the identity which holds for any natural n. (Wikipedia).

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Related pages

Limit of a sequence | Series (mathematics) | Hilbert space | Cauchy–Schwarz inequality | Functional analysis | Mathematics | Inequality (mathematics) | Inner product space | Parseval's identity | Orthonormal basis | Sequence | Parseval's theorem