Functional Analysis

Infinite-dimensional linear algebra: Banach and Hilbert spaces, bounded operators, spectral theory, compact operators, and distributions. The mathematics of quantum mechanics and modern PDEs.

Course Overview

Textbook

Kreyszig, Introductory Functional Analysis with Applications

Chapters

TBD

Sections

TBD

Prerequisites

Real Analysis II, Topology

Target

Quantum mechanics, PDE theory, operator algebras

Status

Planned

Chapters will be scaffolded when this course becomes active.

Why This Course

  • Quantum mechanics — observables are operators on Hilbert space

  • Signal processing — \(L^2\) function spaces, orthogonal decompositions

  • Machine learning — reproducing kernel Hilbert spaces (RKHS) underpin kernel methods

  • Numerical analysis — convergence of approximation methods

  • PDEs — weak solutions, Sobolev spaces, existence/uniqueness theorems

  • The summit — this is where analysis, algebra, and topology converge