, 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
tells us that this series converges
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
, excluding when
is less than 1.