Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element ${\displaystyle x}$ in a Hilbert space with respect to an orthonormal sequence. The inequality is named for F. W. Bessel, who derived a special case of it in 1828.

Conceptually, the inequality is a generalization of the Pythagorean theorem to infinite-dimensional spaces. It states that the "energy" of a vector ${\displaystyle x}$, given by ${\displaystyle \|x\|^{2}}$, is greater than or equal to the sum of the energies of its projections onto a set of perpendicular basis directions. The value ${\displaystyle |\langle x,e_{k}\rangle |^{2}}$ represents the energy contribution along a specific direction ${\displaystyle e_{k}}$, and the inequality guarantees that the sum of these contributions cannot exceed the total energy of ${\displaystyle x}$.

When the orthonormal sequence forms a complete orthonormal basis, Bessel's inequality becomes an equality known as Parseval's identity. This signifies that the sum of the energies of the projections equals the total energy of the vector, meaning no energy is "lost." The inequality is a crucial tool for establishing the convergence of Fourier series and other series expansions in Hilbert spaces.