Applications

Example (Spectral Theory and ODEs). Consider the Sturm-Liouville problem $-u"' + q(x)u = \lambda u$ on $[a, b]$ with boundary conditions $u(a) = u(b) = 0$. The inverse operator $T = (-d^2/dx^2 + q)^{-1}$ is a compact self-adjoint operator on $L^2[a, b]$. By the spectral theorem, the eigenfunctions form an orthonormal basis, and the eigenvalues $\lambda_n \to \infty$.

In quantum mechanics, the state space of a system is a Hilbert space $H$ (typically $L^2(\mathbb{R}^3)$). Observables are self-adjoint operators on $H$. The spectral theorem guarantees that every observable has a spectral decomposition:

$A = \int_{\sigma(A)} \lambda\, dP(\lambda)$

where $P$ is the projection-valued measure associated with $A$.

Example. The position operator $(X\psi)(x) = x\psi(x)$ and momentum operator $(P\psi)(x) = -i\hbar\psi'(x)$ are self-adjoint on suitable domains of $L^2(\mathbb{R})$. The canonical commutation relation $[X, P] = i\hbar I$ is fundamental to quantum mechanics.

Classical solutions of PDEs require pointwise differentiability, which is too restrictive for many problems. Functional analysis provides the framework for weak (distributional) solutions.

Consider the Poisson equation $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$ with $u|_{\partial\Omega} = 0$. Multiply both sides by a smooth test function $\varphi$ with $\varphi|_{\partial\Omega} = 0$ and integrate by parts:

$\int_\Omega \nabla u \cdot \nabla \varphi\, dx = \int_\Omega f \varphi\, dx$

This is the weak formulation: find $u$ in a suitable function space such that the above holds for all test functions $\varphi$.

The natural space for this problem is the Sobolev space $H^1_0(\Omega)$, defined as the completion of $C_c^\infty(\Omega)$ under the norm $\|u\|_{H^1} = \left(\int_\Omega |u|^2 + |\nabla u|^2\, dx\right)^{1/2}$. The Riesz representation theorem applied to the inner product $\langle u, v\rangle_{H^1_0} = \int_\Omega \nabla u \cdot \nabla v\, dx$ yields existence and uniqueness of the weak solution. This is known as the Lax-Milgram theorem:

Theorem 7.1 (Lax-Milgram). Let $H$ be a Hilbert space and $B : H \times H \to \mathbb{R}$ a bounded bilinear form that is coercive, i.e., there exists $c > 0$ such that $B(u, u) \geq c\|u\|^2$ for all $u \in H$. Then for every bounded linear functional $\ell \in H^*$, there exists a unique $u \in H$ such that $B(u, v) = \ell(v)$ for all $v \in H$.

Sobolev spaces $W^{k,p}(\Omega)$ (for $k \geq 0$, $1 \leq p \leq \infty$) generalise this idea: $W^{k,p}$ consists of functions whose weak derivatives up to order $k$ belong to $L^p$. They are Banach spaces, and $W^{k,2} = H^k$ are Hilbert spaces.

The Fourier transform on $L^2(\mathbb{R}^n)$ is a unitary operator $\mathcal{F} : L^2 \to L^2$ defined by:

$(\mathcal{F}f)(\xi) = \frac{1}{(2\pi)^{n/2}} \int_{\mathbb{R}^n} f(x) e^{-i x \cdot \xi} dx$

Plancherel’s theorem states $\|\mathcal{F}f\|_2 = \|f\|_2$. The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of rapidly decaying smooth functions is dense in $L^2$ and is invariant under the Fourier transform.

In signal processing, the sampling theorem (Nyquist-Shannon) follows from the Paley-Wiener theorem characterising functions with compactly supported Fourier transforms. Functional analysis also underpins wavelet theory: the existence of orthonormal wavelet bases of $L^2(\mathbb{R})$ relies on the theory of frames and multiresolution analysis.

The Galerkin method approximates solutions of PDEs by projecting onto finite-dimensional subspaces. Let $V_h \subset H^1_0(\Omega)$ be a finite-dimensional subspace (e.g., finite elements). Find $u_h \in V_h$ such that:

$\int_\Omega \nabla u_h \cdot \nabla v_h\, dx = \int_\Omega f v_h\, dx \quad \forall v_h \in V_h$

This reduces to solving a linear system $A\mathbf{u} = \mathbf{b}$ where $A_{ij} = \int_\Omega \nabla\phi_i \cdot \nabla\phi_j\, dx$ for basis functions $\{\phi_i\}$. Céa’s lemma provides a quasi-optimal error estimate:

$\|u - u_h\|_{H^1} \leq C \inf_{v_h \in V_h} \|u - v_h\|_{H^1}$

In convex optimisation, the existence of minimisers for a functional $J : H \to \mathbb{R}$ follows from the direct method of calculus of variations: if $J$ is coercive, lower semicontinuous, and $H$ is reflexive, then a minimiser exists. This applies to problems such as:

$\min_{u \in H^1_0(\Omega)} \frac{1}{2} \int_\Omega |\nabla u|^2\, dx - \int_\Omega f u\, dx$

The optimality condition is precisely the weak formulation of the Poisson equation.

Theorem 7.2 (Stampacchia). Let $H$ be a Hilbert space, $K \subset H$ a nonempty closed convex set, and $a(\cdot, \cdot)$ a coercive bilinear form. For any $\ell \in H^*$, there exists a unique $u \in K$ such that $a(u, v - u) \geq \ell(v - u)$ for all $v \in K$. This is the foundation of variational inequalities and obstacle problems.

Problem. Solve the Fredholm integral equation $u(x) - \lambda \int_a^b K(x, y) u(y) dy = f(x)$ where $K$ is a continuous kernel on $[a, b]^2$.

Problem. Solve $u - Tu = f$ by iteration where $\|T\| < 1$.

In optimal control, the linear-quadratic regulator (LQR) problem seeks to minimise:

$J(u) = \int_0^\infty (x(t)^T Q x(t) + u(t)^T R u(t))\, dt$

subject to $\dot{x} = Ax + Bu$. The optimal control is $u = -Kx$ where $K = R^{-1}B^T P$ and $P$ solves the algebraic Riccati equation:

$A^T P + P A - P B R^{-1} B^T P + Q = 0$

The existence and uniqueness of a positive definite solution $P$ follows from spectral theory of Hamiltonian matrices and functional analysis on the space of symmetric operators.

A reproducing kernel Hilbert space (RKHS) $H$ is a Hilbert space of functions on $X$ such that the evaluation functional $\delta_x(f) = f(x)$ is bounded for all $x \in X$. By the Riesz representation theorem, there exists $k_x \in H$ with $f(x) = \langle f, k_x \rangle$.

The kernel $k(x, y) = \langle k_x, k_y \rangle$ is a positive definite function. The Moore-Aronszajn theorem states that every positive definite kernel corresponds to a unique RKHS. RKHS theory underpins kernel methods in machine learning, including support vector machines and Gaussian processes.