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.
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 which can be described in terms of potential basis .
For a complete orthonormal sequence (that is, for an orthonormal sequence which 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.
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