Bessel's Inequality

Bessel's inequality bounds the coefficients of a vector relative to an orthonormal family in a real or complex Hilbert space . Use an inner product linear in its first argument, put , and form the finite orthogonal projection

(1)

The orthonormal set gives the residual identity

(2)

Thus every finite partial sum is bounded by , and for a countable orthonormal family,

(3)

For a finite family, equality holds iff lies in its vector space span. For a countable family, equality holds iff lies in the closure of that vector space span (Axler 2026, pp. 241-243). A complete orthogonal system that is normalized therefore gives Parseval's theorem for every . An incomplete family can still give equality for individual vectors in its closed vector space span.

For a weighted generalized Fourier series on an interval , take a nonnegative measurable function and functions satisfying , where is the Kronecker delta. If , the coefficients are

(4)

and Bessel's inequality becomes

(5)

This proof uses only finite orthogonal projections before taking a limit; it does not assume that the generalized Fourier series converges to .

For a square integrable real function on , the usual Fourier series coefficients are for and for . With this normalization,

(6)

For this complete orthogonal system, equality holds by Parseval's theorem.