orthonormal basis
An orthonormal basis is the perfect coordinate system for a Hilbert space: a collection of unit-length vectors, all mutually perpendicular, rich enough that every vector in the space can be reconstructed from its shadows along them. It generalizes the standard axes of R^n, where to find a coordinate you just project onto the corresponding axis — no matrix inversion, no fuss.
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 an orthonormal basis if, in addition, it is complete: the only vector orthogonal to all of them is 0, equivalently their finite linear combinations are dense in H. Then every x in H expands as x = sum over k of <x, e_k> e_k, the series converging in norm, with coefficients given by simple projections.
Crucially, in infinite dimensions an orthonormal basis is NOT a basis in the algebraic (Hamel) sense — you genuinely need the infinite sum, not just finite combinations. Completeness is also equivalent to Parseval's identity ||x||^2 = sum over k of |<x, e_k>|^2, the infinite-dimensional Pythagorean theorem, which says no length is lost when you pass to coordinates. A Hilbert space has a countable orthonormal basis exactly when it is separable.
In l^2 the vectors e_k = (0, ..., 0, 1, 0, ...) with a single 1 in slot k form an orthonormal basis. For x = (x_1, x_2, ...) the coefficients are just <x, e_k> = x_k, the expansion x = sum x_k e_k is the obvious one, and Parseval reads ||x||^2 = sum |x_k|^2 — the very condition for membership in l^2.
The standard basis of l^2, with Parseval's identity recovering the defining norm.
Without completeness the family is merely an orthonormal system, and one only gets Bessel's inequality sum |<x, e_k>|^2 <= ||x||^2, a one-sided bound. Equality for every x is exactly the upgrade to a basis. The Gram–Schmidt process manufactures orthonormal systems from any independent family.