Hilbert Spaces - Key Properties

The geometric structure of Hilbert spaces is fundamentally shaped by the concepts of orthogonality and orthonormal bases, which generalize familiar notions from finite-dimensional Euclidean geometry.

DefinitionOrthogonality

Two vectors $x, y$ in a Hilbert space $H$ are orthogonal, written $x \perp y$ , if $\langle x, y \rangle = 0$ .

For a subset $S \subset H$ , the orthogonal complement is $S^\perp = \{x \in H : \langle x, y \rangle = 0 \text{ for all } y \in S\}$

The orthogonal complement $S^\perp$ is always a closed subspace of $H$ .

DefinitionOrthonormal Sets and Bases

A set $\{e_\alpha\}_{\alpha \in A}$ in $H$ is orthonormal if $\langle e_\alpha, e_\beta \rangle = \delta_{\alpha\beta} = \begin{cases} 1 & \alpha = \beta \\ 0 & \alpha \neq \beta \end{cases}$

An orthonormal set is called an orthonormal basis (or complete orthonormal system) if its closed linear span equals $H$ . Equivalently, $x \perp e_\alpha$ for all $\alpha$ implies $x = 0$ .

ExampleClassical Orthonormal Bases

Standard Basis in $\ell^2$ : The sequence $e_n = (0, \ldots, 0, 1, 0, \ldots)$ with $1$ in the $n$ -th position forms an orthonormal basis
Fourier Basis in $L^2[0, 2\pi]$ : The functions $\frac{1}{\sqrt{2\pi}}, \quad \frac{\cos(nx)}{\sqrt{\pi}}, \quad \frac{\sin(nx)}{\sqrt{\pi}} \quad (n = 1, 2, 3, \ldots)$ form an orthonormal basis
Legendre Polynomials: After normalization, Legendre polynomials form an orthonormal basis for $L^2[-1,1]$

Every element in a Hilbert space can be uniquely represented as a (possibly infinite) linear combination of orthonormal basis vectors. If $\{e_\alpha\}$ is an orthonormal basis, then for any $x \in H$ : $x = \sum_\alpha \langle x, e_\alpha \rangle e_\alpha$ where the sum converges in the norm topology.

TheoremBessel's Inequality and Parseval's Identity

Let $\{e_n\}_{n=1}^\infty$ be an orthonormal sequence in a Hilbert space $H$ . Then for any $x \in H$ :

Bessel's Inequality: $\sum_{n=1}^\infty |\langle x, e_n \rangle|^2 \leq \|x\|^2$

If $\{e_n\}$ is an orthonormal basis, then:

Parseval's Identity: $\sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2$

Remark

Every separable Hilbert space (containing a countable dense subset) has a countable orthonormal basis and is therefore isometrically isomorphic to $\ell^2$ . This remarkable fact means that, from an abstract perspective, there is essentially only one infinite-dimensional separable Hilbert space.

The existence of orthonormal bases and the validity of Parseval's identity are what make Fourier analysis work in Hilbert spaces, providing the foundation for signal processing, quantum mechanics, and harmonic analysis.