Introduction to Hilbert Spaces
A Hilbert space, named after David Hilbert (1862-1943), is a complete inner product space—the natural generalization of Euclidean space to infinite dimensions. These spaces form the mathematical foundation of quantum mechanics, Fourier analysis, signal processing, and many areas of modern mathematics and physics.
While finite-dimensional inner product spaces (like $\mathbb{R}^n$ and $\mathbb{C}^n$) are relatively straightforward, Hilbert spaces allow us to work with infinite sequences, functions, and other infinite-dimensional objects while preserving essential geometric intuitions about length, angle, and orthogonality.
A Hilbert space $H$ is a vector space over the field $\mathbb{R}$ or $\mathbb{C}$ equipped with:
- An inner product $\langle \cdot, \cdot \rangle: H \times H \to \mathbb{F}$ satisfying:
- Conjugate symmetry: $\langle x, y \rangle = \overline{\langle y, x \rangle}$
- Linearity in first argument: $\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$
- Positive definiteness: $\langle x, x \rangle \geq 0$, with equality iff $x = 0$
- A norm induced by the inner product: $\|x\| = \sqrt{\langle x, x \rangle}$
- Completeness with respect to this norm: Every Cauchy sequence in $H$ converges to an element of $H$
The completeness condition distinguishes Hilbert spaces from general inner product spaces and is crucial for many analytical results.
Key Examples of Hilbert Spaces
1. Euclidean Space $\mathbb{R}^n$
The simplest Hilbert space with inner product $\langle x, y \rangle = \sum_{i=1}^n x_i y_i$. Finite-dimensional, complete, with standard basis $\{e_1, \ldots, e_n\}$.
2. Sequence Space $\ell^2$
The space of square-summable sequences: $\ell^2 = \{(x_n)_{n=1}^\infty : \sum_{n=1}^\infty |x_n|^2 < \infty\}$ with inner product $\langle x, y \rangle = \sum_{n=1}^\infty x_n \overline{y_n}$.
3. Function Space $L^2([a,b])$
Square-integrable functions: $L^2([a,b]) = \{f: [a,b] \to \mathbb{C} : \int_a^b |f(x)|^2 dx < \infty\}$ with inner product $\langle f, g \rangle = \int_a^b f(x)\overline{g(x)} dx$.
Hilbert spaces satisfy all properties: vector space structure + inner product + norm + completeness
Fundamental Theorems of Hilbert Space Theory
The Riesz Representation Theorem
Let $H$ be a Hilbert space and $f: H \to \mathbb{F}$ a continuous linear functional. Then there exists a unique vector $y_f \in H$ such that:
Moreover, $\|f\| = \|y_f\|$, establishing an isometric isomorphism between $H$ and its dual space $H^*$.
Key Steps:
- If $f = 0$, take $y_f = 0$.
- Otherwise, let $M = \ker f$, a closed proper subspace of $H$.
- Choose $z \in M^\perp$ with $\|z\| = 1$ (exists by projection theorem).
- For any $x \in H$, write $x = (x - \frac{f(x)}{f(z)}z) + \frac{f(x)}{f(z)}z$.
- The first term is in $M$, the second in $M^\perp$.
- Take $y_f = \overline{f(z)} z$, then verify $f(x) = \langle x, y_f \rangle$.
This theorem is fundamental because it identifies continuous linear functionals with vectors in the space itself.
The Projection Theorem
Let $H$ be a Hilbert space and $C \subseteq H$ a nonempty closed convex subset. Then for every $x \in H$, there exists a unique $P_C(x) \in C$ such that:
If $C$ is a closed subspace, then $P_C$ is linear and $x - P_C(x) \perp C$.
Orthonormal Bases in Hilbert Spaces
Unlike finite dimensions, Hilbert spaces can have infinite orthonormal bases. An orthonormal basis for a Hilbert space $H$ is a set $\{e_\alpha\}_{\alpha \in A}$ such that:
- $\langle e_\alpha, e_\beta \rangle = \delta_{\alpha\beta}$ (orthonormality)
- The linear span is dense in $H$ (completeness)
Every Hilbert space has an orthonormal basis (by Zorn's lemma), and all orthonormal bases have the same cardinality (the dimension of $H$).
Fourier Basis
In $L^2([-\pi, \pi])$, the set $\{\frac{1}{\sqrt{2\pi}} e^{inx}\}_{n \in \mathbb{Z}}$ forms an orthonormal basis. Fourier series expansion: $f(x) = \sum_{n=-\infty}^\infty \hat{f}(n) e^{inx}$.
Legendre Polynomials
In $L^2([-1, 1])$, Legendre polynomials $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$ form an orthogonal basis after normalization.
Hermite Functions
In $L^2(\mathbb{R})$, Hermite functions $\psi_n(x) = H_n(x) e^{-x^2/2}$ form an orthonormal basis important in quantum mechanics.
Spectral Theory in Hilbert Spaces
Spectral theory generalizes the concept of eigenvalues and eigenvectors to infinite-dimensional spaces. For a bounded linear operator $T: H \to H$, we define:
| Concept | Finite Dimension | Hilbert Space |
|---|---|---|
| Eigenvalue | $\lambda$ with $Tv = \lambda v$, $v \neq 0$ | Same, but spectrum may include continuous spectrum |
| Spectrum | Set of eigenvalues $\sigma(T) = \{\lambda_1, \ldots, \lambda_n\}$ | $\sigma(T) = \{\lambda \in \mathbb{C}: T - \lambda I \text{ not invertible}\}$ |
| Spectral Theorem | $T = PDP^{-1}$ with $D$ diagonal | $T = \int_{\sigma(T)} \lambda dE(\lambda)$ with spectral measure $E$ |
Let $H$ be a Hilbert space and $T: H \to H$ a bounded self-adjoint operator ($T = T^*$). Then there exists a unique spectral measure $E$ on $\sigma(T)$ such that:
This allows functional calculus: $f(T) = \int_{\sigma(T)} f(\lambda) dE(\lambda)$ for measurable $f$.
Applications of Hilbert Spaces
In quantum mechanics, the state of a physical system is represented by a vector $\psi$ in a Hilbert space $H$ with $\|\psi\| = 1$ (normalization).
- Observables correspond to self-adjoint operators $A: H \to H$
- Measurement outcomes are eigenvalues of $A$
- Expectation value: $\langle A \rangle_\psi = \langle \psi, A\psi \rangle$
- Time evolution governed by Schrödinger equation: $i\hbar\frac{\partial\psi}{\partial t} = H\psi$
The spectral theorem ensures that measurements yield real numbers and allows probability calculations.
In signal processing, signals are represented as vectors in $L^2(\mathbb{R})$ or $\ell^2(\mathbb{Z})$.
- Fourier transform: Unitary operator $F: L^2(\mathbb{R}) \to L^2(\mathbb{R})$
- Filtering: Projection onto frequency bands
- Compression: Approximation in subspaces spanned by wavelets
- Sampling theorem: Reconstruction from samples using sinc basis
Hilbert space geometry provides optimal approximation methods and error bounds.
Reproducing Kernel Hilbert Spaces (RKHS) are Hilbert spaces of functions where point evaluation is continuous.
- Kernel trick: Map data to high-dimensional feature space
- Support Vector Machines: Find optimal separating hyperplane
- Gaussian Processes: Bayesian nonparametric regression
- Representer theorem: Solutions are linear combinations of kernel evaluations
The RKHS framework provides theoretical foundation for kernel methods in machine learning.
Connection to Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality is fundamental to Hilbert space theory. In any Hilbert space $H$, for all $x, y \in H$:
This inequality has several important consequences in Hilbert spaces:
- Continuity of inner product: If $x_n \to x$ and $y_n \to y$, then $\langle x_n, y_n \rangle \to \langle x, y \rangle$
- Parallelogram law: $\|x + y\|^2 + \|x - y\|^2 = 2(\|x\|^2 + \|y\|^2)$
- Polarization identity (for real spaces): $\langle x, y \rangle = \frac{1}{4}(\|x+y\|^2 - \|x-y\|^2)$
- Angle definition: $\theta = \arccos\left(\frac{\langle x, y \rangle}{\|x\|\|y\|}\right)$ when $\|x\|\|y\| \neq 0$
Solving the Viral Limit Problem in Hilbert Spaces
The viral limit problem from our limit calculator page can be interpreted in a Hilbert space context:
Given sequences $\{h_p\}, \{b_p\}, \{z_p\}$ in a Hilbert space $H$ with:
Assuming $\|h_p\| \to 1$ and using Cauchy-Schwarz, we find $\lim \langle b_p, z_p \rangle = 0.84375$.
This illustrates how Hilbert space techniques solve limit problems involving inner products. For the complete solution, see our dedicated solution page.
Advanced Topics: Unbounded Operators
While bounded operators are well-behaved, many important operators in physics are unbounded (e.g., position and momentum operators in quantum mechanics).
Densely Defined Operators
An operator $T$ with domain $D(T) \subseteq H$ dense in $H$. Essential for defining adjoints and studying spectral properties.
Closed Operators
An operator whose graph $\{(x, Tx): x \in D(T)\}$ is closed in $H \times H$. Many differential operators are closed but unbounded.
Spectral Theory
For unbounded self-adjoint operators, the spectral theorem still holds via projection-valued measures, though technicalities increase.
🚀 Ready to Apply Hilbert Space Theory?
Use our limit calculator to solve inner product limit problems, or study Cauchy-Schwarz inequality for the theoretical foundation.
Historical Context and Modern Developments
Hilbert space theory emerged from David Hilbert's work on integral equations (1904-1910) and was fully developed by John von Neumann in the 1920s to provide rigorous mathematical foundations for quantum mechanics. Key developments include:
- 1907: Frigyes Riesz introduces $L^p$ spaces
- 1910: Ernst Fischer and Frigyes Riesz prove Riesz-Fischer theorem: $L^2$ is complete
- 1929: John von Neumann coins term "Hilbert space" and develops spectral theory
- 1932: von Neumann's "Mathematical Foundations of Quantum Mechanics" establishes Hilbert spaces as framework for quantum theory
- 1950s: Grothendieck develops topological tensor products of Hilbert spaces
- 2000s: Applications expand to machine learning (RKHS theory) and quantum computing
❓ Hilbert Spaces FAQ
What's the difference between Hilbert space and Banach space?
A Hilbert space is a Banach space whose norm comes from an inner product. All Hilbert spaces are Banach spaces, but not conversely (e.g., $L^1$ is Banach but not Hilbert). Hilbert spaces have richer geometric structure (angles, orthogonality).
Are all Hilbert spaces separable?
No. While most applications use separable Hilbert spaces (having countable dense subset), non-separable Hilbert spaces exist. $L^2(\mathbb{R})$ is separable, but $L^2(\mathbb{R}^\mathbb{R})$ with uncountable product measure is not.
How do Hilbert spaces relate to quantum computing?
In quantum computing, qubit states live in finite-dimensional Hilbert spaces $\mathbb{C}^{2^n}$ for $n$ qubits. Quantum gates are unitary operators, and measurement corresponds to projection onto subspaces. Entanglement is modeled via tensor products of Hilbert spaces.
What is a reproducing kernel Hilbert space (RKHS)?
An RKHS is a Hilbert space of functions where point evaluation $f \mapsto f(x)$ is continuous for each $x$. By Riesz theorem, there exists $K_x \in H$ with $f(x) = \langle f, K_x \rangle$. The kernel $K(x,y) = \langle K_x, K_y \rangle$ reproduces function values.