complete orthonormal system

In R^n you expand any vector along an orthonormal basis: v = sum <v,e_k> e_k. A complete orthonormal system is the infinite-dimensional version of that, the engine behind Fourier series. You have infinitely many mutually perpendicular unit vectors, and any vector in the space is the limit of its expansion against them. The catch is the word complete — you need enough of them that nothing nonzero is perpendicular to all of them at once.

Precisely: a family (e_k) in a Hilbert space H is orthonormal if <e_j, e_k> equals 1 when j = k and 0 otherwise. It is complete (or maximal) when the only vector orthogonal to every e_k is the zero vector. Equivalently, every v in H equals sum <v,e_k> e_k, the series converging in norm. The numbers <v,e_k> are the generalized Fourier coefficients, and they capture v completely.

Why completeness is the whole point: Bessel's inequality always gives sum |<v,e_k>|^2 <= ||v||^2 for any orthonormal family. The system is complete exactly when equality holds for every v — that is Parseval's identity, sum |<v,e_k>|^2 = ||v||^2. Parseval says the expansion loses no energy; the coefficient sequence carries the full norm of the vector. This is precisely what makes the coefficient map an isometry onto ell^2.

A warning about the word basis: a complete orthonormal system is NOT a Hamel (algebraic) basis. Vectors are infinite limits of finite combinations, not finite combinations themselves. Infinite-dimensional Hilbert-space expansions live in the topology — convergence in norm — not in pure algebra. Every separable Hilbert space has a countable complete orthonormal system, built by Gram-Schmidt from any dense sequence.