Functional Analysis, Operators & Distributions

Reference notes on infinite-dimensional linear algebra — the theory of vectors, operators, and functionals on function spaces — together with distribution theory. This is the rigorous foundation of quantum theory: the states of QM are vectors in a Hilbert space, observables are self-adjoint operators, measurement is governed by the spectral theorem, dynamics by Stone's theorem, and quantum fields are literally operator-valued tempered distributions.

These pages are companion material to the Mathematics section. They extend the real analysis spine (which stops at metric spaces) into the infinite-dimensional operator theory it does not reach, and supply the rigorous underpinning that QM/preliminaries.md, QM/postulates.md, and QFT/postulates.md assume.

Contents

A. Spaces

1. Normed and Banach Spaces — norms, completeness, $L^p$, bounded operators, dual spaces, and the four "big theorems".
2. Hilbert Spaces — inner products, orthonormal bases, the projection theorem, and Riesz representation; the home of quantum states.

B. Operators

3. Bounded Operators and Their Spectra — adjoints, self-adjoint/unitary/normal/compact operators, and the spectrum.
4. The Spectral Theorem — projection-valued measures, functional calculus, and Stone's theorem $U(t)=e^{-iHt}$.
5. Unbounded Operators and Self-Adjointness — domains, symmetric vs. self-adjoint, deficiency indices, and why $\hat{x},\hat{p}$ are unbounded.

C. Distributions

6. Distributions and the Schwartz Space — test functions, the Dirac delta, distributional derivatives, tempered distributions, and Green's functions.
7. Rigged Hilbert Space (Gel'fand Triples) — the triple $\Phi \subset \mathcal{H}\subset {\Phi }^{\prime }$ that legitimizes continuous spectra and Dirac's $|x⟩$, $|p⟩$ kets.

D. Operator algebras and the bridge

8. C*- and von Neumann Algebras — the algebraic view of observables, the GNS construction, and the Wightman/OS axioms.
9. Why This Matters — The Quantum Bridge — the full dictionary from Hilbert space to the postulates of quantum theory.

Reading order

The spine runs $$\text{normed-banach}\to \text{hilbert-spaces}\to \text{bounded-operators}\to \text{spectral-theorem}\to \text{unbounded-operators}\to \text{distributions},$$ which serves QM and QFT at once; rigged Hilbert space and the physics bridge sit on top, and operator algebras is the algebraic-QFT extension. Prerequisites (metric spaces, completeness, integration) come from the analysis spine; the Fourier transform and Wick-rotation adjacency link to complex analysis. Readers who only need a specific result can jump in anywhere.