In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element
Let
where 〈•,•〉 denotes the inner product in the Hilbert space
consisting of 'infinite sum' of vector resolute
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
Bessel's inequality follows from the identity:
which holds for any natural n.