A Hilbert space is a fundamental concept in mathematics, particularly in functional analysis, that generalizes the notion of Euclidean space. It is a vector space equipped with an inner product that allows for the definition of distance and angle, and crucially, it is complete with respect to the norm induced by this inner product.
Definition
A Hilbert space $\mathcal{H}$ is a real or complex vector space that satisfies the following conditions:
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Inner Product Space: It is endowed with an inner product (also called a scalar product), denoted $\langle x, y \rangle$, which maps any two vectors $x, y \in \mathcal{H}$ to a scalar (real or complex number) and satisfies the following properties:
- Linearity in the first argument: $\langle ax+by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$ for scalars $a, b$ and vectors $x, y, z$.
- Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$ (for real spaces, this simplifies to $\langle x, y \rangle = \langle y, x \rangle$).
- Positive-definiteness: $\langle x, x \rangle \ge 0$, and $\langle x, x \rangle = 0$ if and only if $x = 0$. The inner product induces a norm (or length) by $|x| = \sqrt{\langle x, x \rangle}$.
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Completeness: The space is complete with respect to the metric induced by this norm. This means that every Cauchy sequence of vectors in $\mathcal{H}$ converges to a vector that is also in $\mathcal{H}$.
An inner product space that is not necessarily complete is called a pre-Hilbert space. A Hilbert space is essentially a complete pre-Hilbert space.
Key Properties
- Orthogonality: The inner product allows for the definition of orthogonal vectors: two vectors $x, y$ are orthogonal if $\langle x, y \rangle = 0$. This generalizes the concept of perpendicularity.
- Orthonormal Bases: Every Hilbert space has an orthonormal basis (a set of mutually orthogonal unit vectors), generalizing the concept of coordinate axes. These bases can be finite, countably infinite, or uncountably infinite. For separable Hilbert spaces (those with a countable dense subset), all orthonormal bases have the same cardinality.
- Projection Theorem: For any closed subspace $M$ of a Hilbert space $\mathcal{H}$, any vector $x \in \mathcal{H}$ can be uniquely decomposed into $x = m + n$, where $m \in M$ and $n$ is orthogonal to $M$.
- Riesz Representation Theorem: This theorem establishes a fundamental connection between a Hilbert space and its dual space (the space of continuous linear functionals), stating that every continuous linear functional can be uniquely represented as an inner product with a unique vector in the Hilbert space.
Examples
- Finite-dimensional Euclidean space: $\mathbb{R}^n$ with the standard dot product, or $\mathbb{C}^n$ with the standard complex inner product. These are the most intuitive examples.
- Sequence spaces: The space $l^2(\mathbb{N})$ consists of all sequences of complex numbers $(x_k){k=1}^\infty$ such that $\sum{k=1}^\infty |x_k|^2 < \infty$. The inner product is defined as $\langle x, y \rangle = \sum_{k=1}^\infty x_k \overline{y_k}$.
- Function spaces: The space $L^2([a, b])$ of square-integrable functions on an interval $[a, b]$, where the integral of the absolute square of the function is finite. The inner product is given by $\langle f, g \rangle = \int_a^b f(x)\overline{g(x)} dx$. More generally, $L^2(\Omega, \mu)$ for a general measure space $(\Omega, \mu)$.
Applications
Hilbert spaces are indispensable in many areas of mathematics and physics, providing a robust framework for advanced analysis:
- Quantum Mechanics: The state space of a quantum mechanical system is a Hilbert space. Physical observables are represented by self-adjoint operators on this space, and quantum measurements are described by projections onto subspaces.
- Functional Analysis: They form a central object of study, providing a framework for analyzing linear operators and solving differential and integral equations.
- Fourier Analysis: The theory of Fourier series and Fourier transforms, which decompose functions into sums or integrals of simpler oscillatory functions, is elegantly formulated in Hilbert spaces, especially $L^2$ spaces.
- Signal Processing: Used in the analysis and processing of signals, where signals can be represented as vectors in an appropriate Hilbert space, allowing for operations like filtering and compression.
- Partial Differential Equations: Solutions to many PDEs can be found and analyzed using Hilbert space methods, such as those related to Sobolev spaces, which are themselves types of Hilbert spaces.