Hilbert Space

Hilbert Spaces are complete inner product spaces, and they play a fundamental role in functional analysis, quantum mechanics, and various other mathematical disciplines. They are named after the German mathematician David Hilbert, who extensively developed the theory.

Definition

A Hilbert space is a real or complex inner product space that is also complete with respect to the topology induced by the inner product. In other words, every Cauchy sequence in the space converges to a limit in the space.

The inner product on a Hilbert space $\mathcal{H}$ is a function that takes two vectors $x, y \in \mathcal{H}$ and returns a scalar, denoted $\langle x, y \rangle$ . It satisfies the following properties:

Conjugate Symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
Linearity: $\langle \alpha x + \beta y, z \rangle = \alpha \langle x, z \rangle + \beta \langle y, z \rangle$ for all scalars $\alpha, \beta$
Positivity: $\langle x, x \rangle \geq 0$ , with equality if and only if $$x = 0$$

An inner product space is a complex vector space equipped with an inner product. It's also called a pre-Hilbert space and can contain infinite-dimensional vectors.

Examples

Finite-dimensional Euclidean space $\mathbb{R}^n$ with the standard inner product is a Hilbert space.
The space of square-integrable functions $$L^2([a, b])$$ is a Hilbert space with the inner product defined by $\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} \, dx$ .
The space of continuous functions $$C([a, b])$$ with the inner product defined by $\langle f, g \rangle = \int_a^b f(x) g(x) \, dx$ is a Hilbert space.
The space of square-summable sequences $$l^2$$ with the inner product $\langle x, y \rangle = \sum_{n=1}^\infty x_n \overline{y_n}$ .
The space of square-integrable functions over the real line $L^2(\mathbb{R})$ with the inner product $\langle f, g \rangle = \int_{-\infty}^\infty f(x) \overline{g(x)} \, dx$ .
The Hardy space, consisting of holomorphic functions in the upper half-plane that are square-integrable on the real line, is also a Hilbert space.

Properties

Hilbert spaces have several important properties and structures, including orthogonality, projections, and orthonormal bases.

Orthogonality

Two vectors $x, y \in \mathcal{H}$ are said to be orthogonal if $\langle x, y \rangle = 0$ . A set of vectors is called orthogonal if every pair of distinct vectors in the set is orthogonal. An orthonormal set is an orthogonal set where each vector has norm 1.

Projections

A projection in a Hilbert space is a linear operator $P: \mathcal{H} \to \mathcal{H}$ such that $$P^2 = P$$ . Projections are used to decompose the space into orthogonal components.

Orthonormal Bases

An orthonormal basis for a Hilbert space is a set of vectors that is both orthonormal and complete, meaning that every vector in the space can be expressed as a (possibly infinite) linear combination of the basis vectors.

Orthonormal and Orthogonal Systems

An orthogonal system is a family of nonzero vectors that are orthogonal to each other. If the vectors have a norm of 1, it's called an orthonormal system. Orthogonal systems are linearly independent. Pythagorean formula, Bessel's equality and inequality, and the orthogonal decomposition and Riesz representation theorem can all be represented in terms of orthonormal systems of Hilbert spaces.

As a Category

Hilbert spaces over a field $$K$$ form a symmetric monoidal category $\mathbf{Hilb}_K$ , with morphisms given by bounded linear operators, and a tensor product of Hilbert spaces providing the monoidal structure. Since every morphism $T : V \to W$ in Hilbert spaces admits an adjoint map $T^\ast : W \to V$ , determined by the property that for all $v \in V, w \in W$ we have $\langle w, Tv\rangle = \langle T^\ast w, v\rangle$ , and $(T^\ast)^\ast$ . We call that a dagger structure.

Categorically, the dagger structure means existence of a contravariant endofunctor $\dagger$ , that acts as the identity on objects and squares to the identity. In a dagger category, the notion of unitary ismorphisms makes sense: they are the invertible morphisms $$T$$ such that $T^\dagger = T^{-1}$ .

Applications

Hilbert spaces are used in various fields of mathematics and physics.

Vector Bundles

Main article: Vector bundle

Hilbert spaces play a crucial role in the study of vector bundles, where they provide a natural setting for defining metric tensors. These tensors allow the measurement of lengths and angles, essential in differential geometry. In study of Riemannian manifolds, Hilbert spaces over a field $K = \mathbb{R}$ are used to study the geometry of curved spaces. The inner product structure of Hilbert spaces helps in defining the Riemannian metric, which generalizes the notion of angles and distances to curved spaces.

Information Geometry

Main article: Information geometry

In information geometry, Hilbert spaces are used to study statistical manifolds. These Riemannian manifolds are equipped with a Fischer metric derived from information theory, allowing geometric interpretations of statistical concepts.

Machine Learning

Hilbert spaces are an important concept used in kernel methods in machine learning, such as Support Vector Machines (SVM). The kernel trick implicitly maps data into a high-dimensional Hilbert space, making linear separation possible.

Kernel Hilbert Space

Main article: Kernel Hilbert space

A kernel Hilbert space (KHS) is a Hilbert space associated with a positive definite kernel function. The kernel function is a symmetric function $k: X \times X \rightarrow \mathbb{R}$ that satisfies the Mercer's condition. The space is equipped with an inner product, and the kernel function can be expressed as:

$k(x, y) = \langle \phi(x), \phi(y) \rangle$

Where $\phi: X \rightarrow \mathbb{H}$ is a mapping into the Hilbert space.

Quantum Mechanics

In quantum mechanics, Hilbert spaces are used to describe the state space of quantum systems. The inner product represents probability amplitudes, and operators on the space correspond to physical observables.

Signal Processing

Hilbert spaces play a vital role in signal processing, providing a mathematical framework for understanding and manipulating signals. The structure of Hilbert spaces allows for the representation and analysis of signals in both time and frequency domains. Two significant applications of Hilbert spaces in signal processing are the Fourier Transform and the Wavelet Transform.

Fourier Transform

The Fourier Transform is a mathematical tool that decomposes a signal into its constituent frequencies. In the context of Hilbert spaces, the Fourier Transform can be represented as a linear operator acting on the space. The transformation is given by:
$F(f)(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$
Where $$F(f)$$ is the Fourier Transform of the function $$f$$ in the Hilbert space. The inverse Fourier Transform, which reconstructs the signal from its frequency components, is also defined in the Hilbert space:
$F^{-1}(F(f))(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(f)(\omega) e^{j\omega t} d\omega$
The Parseval's theorem in Hilbert spaces ensures that the energy of the signal is preserved under the Fourier Transform.

Wavelet Transform

Main article: Wavelet, wavelet transform

The Wavelet Transform is another powerful tool in signal processing that provides a multi-resolution analysis of signals. It represents a signal in terms of localized wavelets, and in Hilbert spaces, it can be expressed as:
$W(f)(a, b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} f(t) \psi\left(\frac{t - b}{a}\right) dt$
where $\psi(t)$ is the mother wavelet, and $$a, b$$ are the scale and translation parameters, respectively. The inverse Wavelet Transform in Hilbert spaces is given by:
$W^{-1}(W(f))(t) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W(f)(a, b) \psi\left(\frac{t - b}{a}\right) \frac{da db}{|a|^2}$
The Wavelet Transform's flexibility in analyzing signals at different scales and resolutions makes it particularly useful in applications like image compression and noise reduction.

Functional Analysis

Hilbert spaces are central objects in functional analysis, and they provide a natural setting for studying linear operators, spectral theory, and various other analytical concepts.

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