Functional Analysis
Infinite-dimensional vector spaces, bounded operators, spectral theory, and the math behind PDE.
01Functional Analysis (1): Metric Spaces — Distance, Convergence, and Completeness
From the real line to infinite-dimensional function spaces: why completeness is the dividing line.
02Functional Analysis (2): Normed Spaces and Banach Spaces
Norm axioms, classical examples, equivalence of norms in finite dimensions, completeness and why it matters, Schauder …
03Functional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions
Inner products give infinite-dimensional spaces geometric structure — orthogonality, projections, and the Riesz …
04Functional Analysis (4): Dual Spaces and the Hahn-Banach Theorem — Taming Linear Functionals
The Hahn-Banach theorem guarantees enough continuous linear functionals exist to separate points — the foundation for …
05Functional Analysis (5): Weak and Weak-* Topologies — When Norm Convergence Is Too Strong
Norm topology is too fine for many purposes — weak and weak-* topologies provide the compactness results that make …
06Functional Analysis (6): Bounded Linear Operators and the Big Theorems
The Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem — three consequences of completeness …
07Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions
Compact operators are limits of finite-rank operators and inherit much finite-dimensional spectral behavior — the …
08Functional Analysis (8): Spectral Theory — Decomposing Operators
The spectrum generalizes eigenvalues to infinite dimensions — the spectral theorem for bounded self-adjoint operators …
09Functional Analysis (9): Unbounded Operators — When Boundedness Fails
Closed operators, the distinction between symmetric and self-adjoint, deficiency indices, Friedrichs extension, the …
10Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions
C₀-semigroups provide the abstract framework for evolution equations — the Hille-Yosida theorem characterizes which …
11Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions
Distributions extend the notion of function to handle derivatives that don't exist classically — Sobolev spaces provide …
12Functional Analysis (12): Functional Analysis in Action — PDE and Quantum Mechanics
Lax-Milgram for elliptic PDE, variational methods, quantum observables as self-adjoint operators, and Stone's theorem — …