Functional Analysis: Foundations¶

Job: linear analysis on infinite-dimensional normed spaces — Banach/Hilbert structure, duals, and the “big three” (HB, Baire consequences, Riesz).

Core¶

Definition. Norm $\|\cdot\|$ on $V$ (over $\mathbb{R}$ or $\mathbb{C}$): definite, absolutely homogeneous, triangle. Banach: complete normed. Inner product $\langle\cdot,\cdot\rangle$: Banach norm $\|x\|=\sqrt{\langle x,x\rangle}$ $\Rightarrow$ Hilbert.

Object	Definition
Bounded operator	$T\in\mathcal{L}(X,Y)$: $\\|Tx\\|\le M\\|x\\|$; $\\|T\\|=\sup_{\\|x\\|\le 1}\\|Tx\\|$
Dual $X^*$	$\mathcal{L}(X,\mathbb{K})$; always Banach
Weak topology $\sigma(X,X^*)$	coarsest making all $x^\in X^$ continuous
Weak-$$ $\sigma(X^,X)$	coarsest making evaluations at $x\in X$ continuous
Reflexive	$J:X\to X^{*}$, $Jx(x^)=x^*(x)$, is surjective (hence iso)

Hilbert spaces¶

Cauchy–Schwarz; parallelogram law characterizes inner-product norms. Orthogonal projection onto closed convex sets (esp. closed subspaces): $x=Px+(x-Px)$, $x-Px\perp Y$.

Theorem (Riesz representation, Hilbert). $H^*\cong H$ via $y\mapsto\langle\cdot,y\rangle$ (conjugate-linear in $y$ for $\mathbb{C}$).

Orthonormal bases (Hilbertian): $x=\sum\langle x,e_i\rangle e_i$; Parseval. Separable $\Leftrightarrow$ countable ONB. $L^2$, $\ell^2$ model separable infinite-dim Hilbert spaces (all isomorphic).

Classical Banach spaces¶

Space	Notes
$\ell^p$, $L^p$ ($1\le p<\infty$)	dual $\ell^q$, $L^q$ ($1/p+1/q=1$); reflexive for $1<p<\infty$
$\ell^\infty$, $L^\infty$	duals larger than $\ell^1$, $L^1$
$C(K)$	dual $=$ regular Borel measures (Riesz–Markov)
$c_0$	dual $\ell^1$; not reflexive

The big theorems (Baire package)¶

Assume $X,Y$ Banach.

Theorem (Uniform boundedness / Banach–Steinhaus). Pointwise bounded family $T_\alpha\in\mathcal{L}(X,Y)$ is uniformly bounded in operator norm.

Theorem (Open mapping). Surjective $T\in\mathcal{L}(X,Y)$ is open. Corollary (bounded inverse): bijective bounded linear $\Rightarrow$ inverse bounded.

Theorem (Closed graph). Linear $T:X\to Y$ with closed graph $\{(x,Tx)\}$ is continuous.

Hahn–Banach¶

Theorem (HB, analytic). Dominated extension of linear functionals: $p$ sublinear, $f\le p$ on subspace $\Rightarrow$ extension with $f\le p$. Corollaries: (i) $X^*$ separates points; (ii) $\|x\|=\sup_{\|x^*\|\le 1}|x^*(x)|$; (iii) geometric forms: separate convex sets by hyperplanes (interior hypotheses).

Compact operators and spectra (quals core)¶

$K$ compact: $\overline{K(B_X)}$ compact. Ideal in $\mathcal{L}(X)$. Fredholm alternative for $I-K$: injective $\Leftrightarrow$ surjective; finite-dim ker and coker.

Spectrum $\sigma(T)=\{\lambda:T-\lambda I\text{ not invertible in }\mathcal{L}(X)\}$; nonempty compact for complex Banach algebras (Gelfand). For compact $T$ on inf-dim $X$: $\sigma(T)\subseteq\{0\}\cup\text{eigenvalues}$, eigenvalues $\to 0$.

Spectral theorem (bounded normal / self-adjoint)¶

Let $H$ be complex Hilbert, $T\in\mathcal{L}(H)$.

Continuous functional calculus (normal). If $T$ is normal ($T^*T=TT^*$), there is a unique isometric $*$-homomorphism $C(\sigma(T))\to\mathcal{L}(H)$, $f\mapsto f(T)$, with $\mathrm{id}\mapsto T$ and $1\mapsto I$. For self-adjoint $T$, $\sigma(T)\subset\mathbb{R}$ and $C(\sigma(T),\mathbb{R})$ maps to self-adjoint operators.

Projection-valued measure (resolution of the identity). For self-adjoint (or normal) $T$ there is a unique spectral measure $E$ on Borel sets of $\sigma(T)$ with values in orthogonal projections such that $$ T=\int_{\sigma(T)}\lambda\,dE(\lambda),\qquad f(T)=\int f(\lambda)\,dE(\lambda). $$ Then $\langle E(\cdot)x,x\rangle$ is a positive Borel measure, and $\|f(T)x\|^2=\int |f|^2\,d\|E x\|^2$.

Multiplication operator form. $T$ normal $\Leftrightarrow$ $T$ unitarily equivalent to multiplication by a bounded measurable $m$ on $L^2(\Omega,\mu)$ (possibly direct integral / multiplicity version): $(M_m\psi)(\omega)=m(\omega)\psi(\omega)$, with $\sigma(T)=\operatorname{ess\,range}(m)$.

Case	Structure
Compact self-adjoint	$Tx=\sum_n \lambda_n\langle x,e_n\rangle e_n$; $\lambda_n\to 0$; ON eigenbasis of $\overline{\operatorname{ran} T}$
Bounded self-adjoint	$E(\cdot)$ on $\mathbb{R}$; support $=\sigma(T)$
Unitary	spectral measure on $S^1$ (Stone / SNAG in abelian groups)

Unbounded operators (pointer)¶

Many differential operators are unbounded: define $T:\operatorname{Dom}(T)\subset H\to H$ densely defined. Closed if graph is closed; adjoint $T^*$ via $\langle Tx,y\rangle=\langle x,T^*y\rangle$ on maximal domain. Self-adjoint: $T=T^*$ (not merely symmetric $T\subset T^*$).

Spectral theorem (unbounded self-adjoint). Same projection-valued measure picture: $T=\int_{\mathbb{R}}\lambda\,dE(\lambda)$ on $\operatorname{Dom}(T)=\{x:\int\lambda^2\,d\|Ex\|^2<\infty\}$. Stone’s theorem: strongly continuous unitary groups $U(t)=e^{itA}$ $\Leftrightarrow$ self-adjoint generator $A$.

Essentially self-adjoint: unique self-adjoint extension of a symmetric operator — required before interpreting quantum observables / Schrödinger operators. Deficiency indices measure failure. See PDE / quantum notes for $-\Delta$ on domains and boundary conditions.

See it¶

PDE weak solutions. Lax–Milgram: continuous coercive sesquilinear form $a$ on Hilbert $V$ $\Rightarrow$ unique $u$ with $a(u,v)=\langle f,v\rangle$. Sobolev spaces $H^1_0$ are the Hilbert setting for elliptic boundary value problems.

Weak compactness. Reflexive: closed balls weakly compact (Eberlein–Šmulian). Used to extract weakly convergent minimizing sequences in calculus of variations.

Check¶

Proof idea: open mapping from Baire.

$Y=\bigcup T(nB_X)$; some $\overline{T(nB)}$ has interior; scale and absorb to show $T(B)$ contains a ball; then a completeness/iteration argument upgrades to open at $0$.

Counterexample: discontinuous linear functional on a normed space.

Needs Hamel basis (AC): define arbitrarily on basis and extend. No constructive example on separable Banach without additional principles; all such functionals are unbounded and non-measurable relative to Borel structure.

Pitfalls¶

Finite-dim: all norms equiv, all linear maps continuous, weak=strong — all fail in infinite dim.
Weak convergence: $x_n\rightharpoonup x$ does not imply $\|x_n\|\to\|x\|$ (only liminf $\ge\|x\|$); uniform convexity upgrades.
$X^*$ separates points via HB — without some form of AC/HB, pathological.
Compact $\neq$ finite rank, but finite-rank are dense among compact on Hilbert (not on all Banach).
Symmetric $\neq$ self-adjoint for unbounded operators; domains are part of the data.
Spectrum of unbounded $T$ is defined via $T-\lambda I$ failing to be bijective $\operatorname{Dom}(T)\to H$ with bounded inverse; residual spectrum pathologies differ from the bounded case.

Refs¶

Rudin, Functional Analysis
Conway, A Course in Functional Analysis
Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations

Next¶

Measure theory · Metric spaces · Real analysis · Fourier analysis · PDE foundations · Topology overview · Condensed mathematics — condensed/liquid re-architects topological vector spaces · ML mathematics — Barron spaces, NTK as RKHS, function spaces for approximation theory · Elliptic PDE — Schauder / Sobolev / Lax–Milgram as functional analysis · SCV / $\bar\partial$-Neumann — the $\bar\partial$-Neumann as unbounded-operator / Sobolev PDE · FA index · Operator Algebras · Spectral Theory · Function Spaces · Harmonic Analysis

Object	Definition
Bounded operator	\(T\in\mathcal{L}(X,Y)\): \(\\|Tx\\|\le M\\|x\\|\); \(\\|T\\|=\sup_{\\|x\\|\le 1}\\|Tx\\|\)
Dual \(X^*\)	\(\mathcal{L}(X,\mathbb{K})\); always Banach
Weak topology \(\sigma(X,X^*)\)	coarsest making all \(x^\in X^\) continuous
Weak-\(\) \(\sigma(X^,X)\)	coarsest making evaluations at \(x\in X\) continuous
Reflexive	\(J:X\to X^{*}\), \(Jx(x^)=x^*(x)\), is surjective (hence iso)

Space	Notes
\(\ell^p\), \(L^p\) (\(1\le p<\infty\))	dual \(\ell^q\), \(L^q\) (\(1/p+1/q=1\)); reflexive for \(1<p<\infty\)
\(\ell^\infty\), \(L^\infty\)	duals larger than \(\ell^1\), \(L^1\)
\(C(K)\)	dual \(=\) regular Borel measures (Riesz–Markov)
\(c_0\)	dual \(\ell^1\); not reflexive