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 .
Bessel's inequality follows from the identity:
which holds for any natural n.
- Hazewinkel, Michiel, ed. (2001), "Bessel inequality", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Bessel's Inequality the article on Bessel's Inequality on MathWorld.