A [vector space](/page/Vector%20Space) with an [inner product](/page/Inner%20Product) gives two different ways to understand a vector. We can treat the vector as a single object, or we can ask how much of it points in each chosen direction. In finite-dimensional Euclidean space this second description feels automatic: choose perpendicular unit coordinate axes, take dot products, and recover the vector from its coordinates. The question behind an orthonormal basis is whether the same idea survives in an infinite-dimensional Hilbert space, where a vector may require infinitely many coordinates and convergence becomes part of the coordinate system itself.

The first warning is that algebraic bases are too rigid for analysis. A function in $L^2(0,1)$ is not usefully described by a finite list of basis vectors, and a Hamel basis gives no workable way to approximate, compute energy, or pass to limits. What analysis needs is a coordinate system compatible with the norm: finite partial sums should approximate the vector, squared coefficients should measure squared length, and orthogonality should turn geometry into scalar arithmetic.

[example: Fourier Sines as Coordinates] Let $\tau>0$ and let $H=L^2(0,\tau)$ with inner product, where $\mathcal L^1$ denotes one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure), \begin{align*} (f,g)_H = \int_0^\tau f(x)\overline{g(x)}\,d\mathcal L^1(x). \end{align*} For $n\in\mathbb N$, define \begin{align*} e_n(x)=\sqrt{\frac{2}{\tau}}\sin\left(\frac{n\pi x}{\tau}\right). \end{align*} For $m\ne n$, the product-to-sum identity gives \begin{align*} (e_m,e_n)_H &=\frac{2}{\tau}\int_0^\tau \sin\left(\frac{m\pi x}{\tau}\right) \sin\left(\frac{n\pi x}{\tau}\right)\,d\mathcal L^1(x)\\ &=\frac{1}{\tau}\int_0^\tau \left[ \cos\left(\frac{(m-n)\pi x}{\tau}\right) - \cos\left(\frac{(m+n)\pi x}{\tau}\right) \right]\,d\mathcal L^1(x)\\ &=\frac{1}{\tau} \left[ \frac{\tau}{(m-n)\pi}\sin\left(\frac{(m-n)\pi x}{\tau}\right) - \frac{\tau}{(m+n)\pi}\sin\left(\frac{(m+n)\pi x}{\tau}\right) \right]_{0}^{\tau}\\ &=\frac{1}{\tau} \left[ \frac{\tau}{(m-n)\pi}\sin((m-n)\pi) - \frac{\tau}{(m+n)\pi}\sin((m+n)\pi) \right]\\ &=0. \end{align*} For $n=m$, the identity $\sin^2\theta=(1-\cos(2\theta))/2$ gives \begin{align*} (e_n,e_n)_H &=\frac{2}{\tau}\int_0^\tau \sin^2\left(\frac{n\pi x}{\tau}\right)\,d\mathcal L^1(x)\\ &=\frac{1}{\tau}\int_0^\tau \left[ 1-\cos\left(\frac{2n\pi x}{\tau}\right) \right]\,d\mathcal L^1(x)\\ &=\frac{1}{\tau} \left[ x-\frac{\tau}{2n\pi}\sin\left(\frac{2n\pi x}{\tau}\right) \right]_{0}^{\tau}\\ &=\frac{1}{\tau}\left[\tau-\frac{\tau}{2n\pi}\sin(2n\pi)\right]\\ &=1. \end{align*} Thus $(e_n)_{n\in\mathbb N}$ is an orthonormal sequence. Now take $f(x)=x$. Its $n$th coefficient is \begin{align*} (f,e_n)_H &=\sqrt{\frac{2}{\tau}}\int_0^\tau x\sin\left(\frac{n\pi x}{\tau}\right)\,d\mathcal L^1(x). \end{align*} Using [integration by parts](/theorems/2098) with $u=x$ and $dv=\sin(n\pi x/\tau)\,dx$, so that $du=dx$ and $v=-\tau\cos(n\pi x/\tau)/(n\pi)$, we get \begin{align*} \int_0^\tau x\sin\left(\frac{n\pi x}{\tau}\right)\,d\mathcal L^1(x) &= \left[ -\frac{\tau x}{n\pi}\cos\left(\frac{n\pi x}{\tau}\right) \right]_0^\tau + \frac{\tau}{n\pi}\int_0^\tau \cos\left(\frac{n\pi x}{\tau}\right)\,d\mathcal L^1(x)\\ &= -\frac{\tau^2}{n\pi}\cos(n\pi) + \frac{\tau}{n\pi} \left[ \frac{\tau}{n\pi}\sin\left(\frac{n\pi x}{\tau}\right) \right]_0^\tau\\ &= -\frac{\tau^2}{n\pi}(-1)^n + \frac{\tau^2}{n^2\pi^2}\sin(n\pi)\\ &= (-1)^{n+1}\frac{\tau^2}{n\pi}. \end{align*} Therefore \begin{align*} (f,e_n)_H &=\sqrt{\frac{2}{\tau}}\cdot (-1)^{n+1}\frac{\tau^2}{n\pi}\\ &=(-1)^{n+1}\sqrt{2}\,\frac{\tau^{3/2}}{n\pi}. \end{align*} The partial sums $\sum_{n=1}^N (f,e_n)_H e_n$ are not algebraic coordinate expansions in a finite-dimensional space; they are approximations in the $L^2$ norm. This example shows the central pattern: orthogonality computes the coefficients explicitly, while completeness decides whether the infinite expansion returns the original vector. [/example]

The purpose of this chapter is to build that pattern in abstract Hilbert spaces. We start with orthonormal systems, separate orthogonality from completeness, prove the energy inequalities that make coefficients meaningful, and then explain why every Hilbert space has enough orthonormal vectors to act as coordinates.

## Orthogonality, Span, and Completeness

The basic obstruction to usable coordinates is interaction between basis vectors. If two directions are not perpendicular, changing one coordinate changes the contribution of another. The inner product removes this interference when the chosen vectors are pairwise orthogonal and normalized to length $1$.

[definition: Orthonormal Set] Let $H$ be a Hilbert space over $\mathbb R$ or $\mathbb C$. A subset $E \subset H$ is an orthonormal set if every $e \in E$ satisfies $\|e\|_H = 1$ and every pair of distinct elements $e,f \in E$ satisfies $(e,f)_H = 0$. [/definition]

Orthonormality is a metric condition, not yet a spanning condition. A single unit vector in an infinite-dimensional Hilbert space is orthonormal, but it does not describe most vectors. The next issue is whether the chosen directions leave a hidden perpendicular direction behind; if they do, every coefficient in the system can vanish while the vector itself remains nonzero.

[definition: Complete Orthonormal Set] Let $H$ be a Hilbert space and let $E \subset H$ be an orthonormal set. The set $E$ is complete if the only vector $x \in H$ satisfying $(x,e)_H = 0$ for every $e \in E$ is $x=0$. [/definition]

Completeness is often the right conceptual condition, but approximation is the condition used in estimates. To connect them, we need the closed subspace generated by a family of vectors. This construction records not only finite linear combinations, but also every vector that can be reached as a norm limit of such combinations.

[definition: Closed Linear Span] Let $H$ be a Hilbert space and let $A \subset H$. The closed linear span of $A$ is \begin{align*} \overline{\operatorname{span}}(A) = \overline{\left\{\sum_{i=1}^n \alpha_i a_i : n \in \mathbb N,\ \alpha_i \in \mathbb F,\ a_i \in A\right\}}, \end{align*} where $\mathbb F$ is the scalar field of $H$ and the closure is taken in the norm of $H$. [/definition]

## Definition

The problem is now to name the kind of orthonormal set that actually serves as coordinates for the whole Hilbert space. Orthogonality alone gives independent directions, while closed spanning gives approximation of arbitrary vectors. The central object of the chapter is the structure that has both features at once.

[definition: Orthonormal Basis] Let $H$ be a Hilbert space. An orthonormal basis of $H$ is a complete orthonormal set $E \subset H$. [/definition]

This terminology differs from the algebraic notion of a Hamel basis. An orthonormal basis usually does not mean that each vector is a finite linear combination of basis elements. It means that each vector is a norm limit of finite linear combinations, with coefficients controlled by the inner product.

## Coefficients

A coordinate system needs a way to extract coordinates without solving a new approximation problem each time. Orthogonality makes this possible: when all other basis directions are perpendicular to $e$, the inner product with $e$ isolates exactly the scalar multiplying that direction. This scalar is the coordinate attached to $e$.

[definition: Fourier Coefficient] Let $H$ be a Hilbert space, let $E \subset H$ be an orthonormal set, and let $x \in H$. For $e \in E$, the Fourier coefficient of $x$ at $e$ is the scalar $(x,e)_H$. [/definition]

The word Fourier here is structural rather than limited to trigonometric functions. It signals that vectors are being analyzed by their projections onto orthonormal directions.

[remark: Linear Convention] Throughout this page the inner product $(\cdot,\cdot)_H$ is linear in the first argument and conjugate-linear in the second argument. With this convention, the coefficient multiplying $e$ in an expansion of $x$ is $(x,e)_H$. [/remark]