Orthonormal bases turn Hilbert-space geometry into coordinates. The point is not merely to expand vectors, but to make norm, convergence, and operators computable through square-summable coefficients.
Orthonormal systems
Orthonormal systems give mutually independent directions in Hilbert space. Bessel's inequality is the first sign that coefficients are square-summable.
Definition. A family \((e_i)_{i\in I}\) in a Hilbert space \(H\) is orthonormal if \(\langle e_i,e_j\rangle=\delta_{ij}\). It is complete if the only vector orthogonal to every \(e_i\) is \(0\).
Theorem. Bessel inequality: for every finite \(F\subset I\), \(\sum_{i\in F}|\langle x,e_i\rangle|^2\le\|x\|^2\).
Proof. Let \(s_F=\sum_{i\in F}\langle x,e_i\rangle e_i\). Orthogonality gives \(\|s_F\|^2=\sum_{i\in F}|\langle x,e_i\rangle|^2\), and \(x-s_F\perp s_F\). Hence \(\|x\|^2=\|x-s_F\|^2+\|s_F\|^2\ge\|s_F\|^2\).
Parseval identity
Parseval's identity is the exact coordinate formula for the norm. It holds precisely when the orthonormal system is complete.
Theorem. For an orthonormal family \((e_i)\), the following are equivalent: it is complete; \(\overline{\operatorname{span}}\{e_i\}=H\); for every \(x\in H\), \(x=\sum_i\langle x,e_i\rangle e_i\) in norm; for every \(x\), \(\|x\|^2=\sum_i|\langle x,e_i\rangle|^2\).
Proof. The orthogonal complement of \(\overline{\operatorname{span}}\{e_i\}\) is exactly the set of vectors orthogonal to all \(e_i\), so the first two conditions are equivalent. If the span is dense, finite orthogonal projections onto spans of finite subfamilies approximate \(x\); Bessel inequality makes the net of partial sums Cauchy, and its limit must be \(x\). Then Pythagoras gives Parseval.
If Parseval holds and \(x\) is orthogonal to every \(e_i\), then \(\|x\|^2=0\), so \(x=0\); hence the family is complete. Norm convergence of the Fourier series also implies Parseval by passing finite-dimensional Pythagoras to the limit.
Existence of orthonormal bases
Every Hilbert space has an orthonormal basis in the Hilbert sense. The proof is a maximality argument plus the projection theorem.
Theorem. Every Hilbert space has a maximal orthonormal basis. If \(H\) is separable, it has a countable orthonormal basis.
Proof. Partially order orthonormal families by inclusion. The union of a chain is orthonormal, so Zorn's lemma gives a maximal orthonormal family. If its closed span were not all of \(H\), choose a nonzero vector orthogonal to that span, normalize it, and enlarge the family, contradicting maximality.
If \(H\) is separable, a disjoint-ball argument shows every orthonormal family is countable: balls \(B(e_i,1/3)\) are pairwise disjoint, and each contains a distinct point from a fixed countable dense set.
Fourier coefficients
Fourier coefficients are coordinates against an orthonormal system. They turn convergence questions into statements about \(\ell^2\) sequences.
Proposition. Let \((e_n)\) be an orthonormal basis. The coefficient map \(U:H\to\ell^2\), \(Ux=(\langle x,e_n\rangle)\), is a surjective linear isometry.
Proof. Parseval gives \(\|Ux\|_{\ell^2}=\|x\|\). For \(a=(a_n)\in\ell^2\), the partial sums \(\sum_{n\le N}a_ne_n\) are Cauchy because their squared increments are \(\sum_{M<n\le N}|a_n|^2\). Let \(x\) be the limit. Then \(\langle x,e_n\rangle=a_n\) by continuity of the inner product.
Classical trigonometric basis
The trigonometric system is the model example linking Hilbert-space expansion with classical Fourier analysis.
Example. In \(L^2[-\pi,\pi]\), the functions \(e_n(t)=(2\pi)^{-1/2}e^{int}\), \(n\in\mathbb Z\), form an orthonormal basis. Completeness is a theorem of Fourier analysis: trigonometric polynomials are dense in \(L^2\). Parseval becomes \(\|f\|_2^2=\sum_{n\in\mathbb Z}|\hat f(n)|^2\).