Functional Analysis

Normed Spaces

Definition: Pair $(X, \|\cdot\|)$ where $X$ is a vector space and $\|\cdot\|: X \to \mathbb{R}_{\geq 0}$ satisfies:

Positivity: $\|x\| = 0 \iff x = 0$

Homogeneity: $\|\alpha x\| = |\alpha| \|x\|$

Triangle: $\|x + y\| \leq \|x\| + \|y\|$

Examples

$L^p$ spaces: $\|f\|_p = \left(\int_\Omega |f|^p\right)^{1/p}$

$\ell^p$ sequences: $\|x\|_p = \left(\sum_{i=1}^\infty |x_i|^p\right)^{1/p}$

$C[a,b]$: $\|f\| = \max_{x \in [a,b]} |f(x)|$

A norm induces a metric $d(x,y) = \|x-y\|$

Banach Spaces

Definition: A complete normed vector space (every Cauchy sequence converges).

Completeness Condition

$\forall \varepsilon > 0, \exists N: \|x_m - x_n\| < \varepsilon$ for $m,n > N$

$\implies \exists x \in X: \lim_{n \to \infty} x_n = x$

Examples

All $L^p(\Omega)$ for $1 \leq p \leq \infty$

All $\ell^p$ for $1 \leq p \leq \infty$

$C[a,b]$ with sup norm

Completeness is what makes limit arguments work

Inner Product Spaces

Definition: $(X, \langle \cdot, \cdot \rangle)$ with $\langle \cdot, \cdot \rangle: X \times X \to \mathbb{F}$ satisfying:

$\langle x, x \rangle \geq 0$ and $= 0 \iff x = 0$

$\langle x, y \rangle = \overline{\langle y, x \rangle}$ (conjugate symmetry)

$\langle \alpha x + \beta y, z \rangle = \alpha \langle x, z \rangle + \beta \langle y, z \rangle$

Induced Norm

$\|x\| = \sqrt{\langle x, x \rangle}$

Cauchy-Schwarz

$|\langle x, y \rangle| \leq \|x\| \|y\|$

Equality iff $x, y$ are linearly dependent

Hilbert Spaces

Definition: A complete inner product space.

Key Properties

Riesz representation: $\forall f \in X^*, \exists! y: f(x) = \langle x, y \rangle$

Orthogonal decomposition: $X = M \oplus M^\perp$ for closed subspace $M$

An orthonormal basis exists

Parseval Identity

$\|x\|^2 = \sum_{i=1}^\infty |\langle x, e_i \rangle|^2$

Bessel Inequality

$\sum_{i=1}^\infty |\langle x, e_i \rangle|^2 \leq \|x\|^2$ (any orthonormal set)

Examples: $L^2(\Omega)$, $\ell^2$

Bounded Linear Operators

Definition: $T: X \to Y$ linear with $\exists C: \|Tx\| \leq C\|x\|$

Equivalences

Bounded $\iff$ continuous

Continuous at $0$ $\iff$ continuous everywhere

Space of Bounded Operators

$B(X,Y) = \{T: X \to Y \text{ bounded}\}$

$B(X,Y)$ with the operator norm is a Banach space if $Y$ is complete

Operator Norms

Definition

$\|T\| = \sup_{\|x\| \leq 1} \|Tx\|$

$= \sup_{x \neq 0} \dfrac{\|Tx\|}{\|x\|}$

$= \inf\{C: \|Tx\| \leq C\|x\| \ \forall x\}$

Properties

$\|T_1 \circ T_2\| \leq \|T_1\| \|T_2\|$ (submultiplicative)

$\|Tx\| \leq \|T\| \|x\|$ for all $x$

Operator norm makes $B(X)$ a Banach algebra

Compact Operators

Definition: $T: X \to Y$ is compact if it maps bounded sets to relatively compact sets.

Equivalences

Bounded sequences map to sequences with convergent subsequences

Approximable by finite-rank operators (on a Hilbert space)

Properties

Compact $\implies$ bounded (hence continuous)

Composition with any bounded operator is compact

Compact operators are the "nice" infinite-dimensional analogue of matrices

Self-Adjoint Operators

Adjoint (Hilbert space)

$\langle Tx, y \rangle = \langle x, T^*y \rangle$

Self-Adjoint

$T = T^*$

Properties

$\langle Tx, x \rangle \in \mathbb{R}$ for all $x$

$\sigma(T) \subseteq \mathbb{R}$ (real spectrum)

Eigenvectors for distinct eigenvalues are orthogonal

Self-adjoint = real-symmetric matrices generalized

Dual Spaces & Functionals

Dual Space

$X^* = B(X, \mathbb{F})$ (bounded linear functionals)

Functional Norm

$\|f\|_{X^*} = \sup_{\|x\| \leq 1} |f(x)|$

Duality Pairing

$\langle f, x \rangle = f(x)$

Reflexivity

Canonical embedding $J: X \to X^{**}$, $\ J(x)(f) = f(x)$

$X$ reflexive if $J$ is surjective (i.e. $X = X^{**}$)

Reflexive: $L^p, \ell^p$ for $1 < p < \infty$

Non-reflexive: $L^1$, $c_0$, $C[a,b]$

$X^*$ is always a Banach space (even if $X$ is incomplete)

Weak & Weak* Convergence

Weak Convergence in $X$

$x_n \rightharpoonup x: \ f(x_n) \to f(x)$ for all $f \in X^*$

Weak* Convergence in $X^*$

$f_n \overset{*}{\rightharpoonup} f: \ f_n(x) \to f(x)$ for all $x \in X$

Properties

Strong convergence $\implies$ weak convergence

Weak (and weak*) limits are unique

Banach-Alaoglu: the closed unit ball of $X^*$ is weak*-compact

Weak topologies recover compactness in infinite dimensions

Spectral Theory Basics

Spectrum

$\sigma(T) = \{\lambda \in \mathbb{C}: (T - \lambda I) \text{ not bijective}\}$

Resolvent Set

$\rho(T) = \mathbb{C} \setminus \sigma(T)$

Resolvent Operator

For $\lambda \in \rho(T)$: $\ R(\lambda, T) = (T - \lambda I)^{-1}$

Facts

$\sigma(T)$ is compact and nonempty (bounded $T$)

Spectral radius: $r(T) = \sup_{\lambda \in \sigma(T)} |\lambda| \leq \|T\|$

Eigenvalues are the point spectrum; the spectrum can be larger

Hahn-Banach Theorem

Analytic form: If $f: M \to \mathbb{F}$ is bounded linear on a subspace $M \subseteq X$, then $\exists F \in X^*$ with:

$F|_M = f$ and $\|F\|_{X^*} = \|f\|_M$

Consequences

$X^*$ separates points: $x \neq 0 \implies \exists f \in X^*: f(x) \neq 0$

$\|x\| = \sup_{f \in X^*, \|f\| \leq 1} |f(x)|$

Guarantees the dual space is rich enough to be useful

Open Mapping & Closed Graph

Open Mapping Theorem

If $T: X \to Y$ is a surjective bounded linear map between Banach spaces, then $T$ is open

Bounded inverse: if $T$ is also bijective, then $T^{-1}$ is bounded ($\|T^{-1}\| < \infty$)

Closed Graph Theorem

$\text{Graph}(T) = \{(x, Tx) : x \in X\} \subseteq X \times Y$

If $T: X \to Y$ is linear between Banach spaces with closed graph, then $T$ is bounded

Criterion: $x_n \to x$ and $Tx_n \to y \implies Tx = y$

Uniform Boundedness

Banach-Steinhaus: Let $\{T_i\}_{i \in I}$ be a family of bounded operators $T_i: X \to Y$ with $X$ Banach.

If $\sup_i \|T_i x\| < \infty$ for each $x \in X$, then:

$\sup_i \|T_i\| < \infty$

Pointwise boundedness $\implies$ uniform boundedness

Spectral Theorem (Compact)

For compact self-adjoint $T$ on a Hilbert space $H$: there is an orthonormal basis $\{e_n\}$ of eigenvectors with real eigenvalues $\lambda_n \to 0$:

$Tx = \sum_n \lambda_n \langle x, e_n \rangle e_n$

Diagonalization: in the eigenbasis, $T$ is diagonal

Generalizes matrix diagonalization to operators

Fredholm Alternative

For a compact operator $T$ on a Banach space $X$: exactly one holds.

Case 1: $(I - T)x = y$ has a unique solution for every $y$; equivalently $\dim \ker(I - T) = 0$ and $(I-T)$ is surjective

Case 2: $(I - T)x = 0$ has a non-trivial solution, and $\dim \ker(I - T) = \dim \ker(I - T^*)$ (finite)

Fundamental for solvability of operator equations

Key Inequalities

Cauchy-Schwarz

$|\langle x, y \rangle| \leq \|x\| \|y\|$

Hölder ($\frac{1}{p}+\frac{1}{q}=1$)

$\left|\int fg\right| \leq \|f\|_p \|g\|_q$

Minkowski (Triangle in $L^p$)

$\|f+g\|_p \leq \|f\|_p + \|g\|_p$

Bessel / Parseval

$\sum_i |\langle x, e_i \rangle|^2 \leq \|x\|^2$ (Bessel)

$\|x\|^2 = \sum_i |\langle x, e_i \rangle|^2$ (Parseval, complete basis)

Cauchy-Schwarz is the $p=q=2$ case of Hölder

Sobolev Spaces

Definition $W^{k,p}(\Omega)$

$W^{k,p}(\Omega) = \{u \in L^p(\Omega): D^\alpha u \in L^p \text{ for } |\alpha| \leq k\}$

Norm

$\|u\|_{W^{k,p}} = \left(\sum_{|\alpha| \leq k} \|D^\alpha u\|_p^p\right)^{1/p}$

Hilbert Case

$H^k(\Omega) = W^{k,2}(\Omega)$ (common for PDEs)

$H^1_0(\Omega)$: functions vanishing on $\partial\Omega$

$D^\alpha$ denotes a weak (distributional) derivative

Applications — PDEs

Weak Formulation

Find $u \in H^1_0(\Omega)$ such that:

$\int_\Omega \nabla u \cdot \nabla v = \int_\Omega fv \quad \forall v \in H^1_0(\Omega)$

Lax-Milgram Theorem

If $a(u,v)$ is bilinear, bounded, and coercive, then a unique $u$ exists

Poisson Equation

$-\Delta u = f \text{ with } u|_{\partial \Omega} = 0$

Functional analysis provides existence + uniqueness of weak solutions

Applications — Quantum Mechanics

Hilbert Space Framework

States live in $H = L^2(\mathbb{R}^3)$

Observables

Self-adjoint operators (position, momentum, energy):

$\hat{x} = x, \quad \hat{p} = -i\hbar\nabla, \quad \hat{H} = \dfrac{\hat{p}^2}{2m} + V(\hat{x})$

Eigenvalue Equation

$\hat{H}\psi = E\psi$ (time-independent Schrödinger)

Spectral theory determines energy levels and bound states