Functional Analysis
Dec 4, 2021
Kernel Methods
44 min readKernel Methods (3): RKHS — The Theoretical Soul of Kernel Methods
Reproducing Kernel Hilbert Space — the function space where kernel methods live. The reproducing property, the representer theorem, and why finite-data optimization works in infinite dimensions.
Nov 29, 2021
Kernel Methods
76 min readKernel Methods (2): Mathematical Foundations — Positive-Definite Kernels and Mercer's Theorem
What makes a function a valid kernel? Positive-definiteness, the Gram matrix test, and Mercer's theorem — the spectral decomposition that justifies the kernel trick.
Oct 23, 2021
Functional Analysis
70 min readFunctional 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 — where the abstract theory meets concrete applications.
Oct 21, 2021
Functional Analysis
72 min readFunctional 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 the right setting for weak solutions to PDE.
Oct 19, 2021
Functional Analysis
76 min readFunctional 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 operators generate well-posed dynamics.
Oct 17, 2021
Functional Analysis
68 min readFunctional Analysis (9): Unbounded Operators — When Boundedness Fails
Closed operators, the distinction between symmetric and self-adjoint, deficiency indices, Friedrichs extension, the spectral theorem for unbounded self-adjoint operators, and Stone's theorem.
Oct 15, 2021
Functional Analysis
68 min readFunctional Analysis (8): Spectral Theory — Decomposing Operators
The spectrum generalizes eigenvalues to infinite dimensions — the spectral theorem for bounded self-adjoint operators and continuous functional calculus give us a complete decomposition.
Oct 13, 2021
Functional Analysis
62 min readFunctional 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 Fredholm alternative and spectral theorem for compact self-adjoint operators.
Oct 11, 2021
Functional Analysis
72 min readFunctional Analysis (6): Bounded Linear Operators and the Big Theorems
The Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem — three consequences of completeness that constrain how operators can behave.
Oct 9, 2021
Functional Analysis
68 min readFunctional 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 optimization and PDE theory work.
Oct 7, 2021
Functional Analysis
76 min readFunctional 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 duality theory in functional analysis.
Oct 5, 2021
Functional Analysis
68 min readFunctional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions
Inner products give infinite-dimensional spaces geometric structure — orthogonality, projections, and the Riesz representation theorem make Hilbert spaces the analyst's paradise.
Oct 3, 2021
Functional Analysis
78 min readFunctional Analysis (2): Normed Spaces and Banach Spaces
Norm axioms, classical examples, equivalence of norms in finite dimensions, completeness and why it matters, Schauder bases, quotient spaces, and the role of separability.
Oct 1, 2021
Functional Analysis
70 min readFunctional Analysis (1): Metric Spaces — Distance, Convergence, and Completeness
From the real line to infinite-dimensional function spaces: why completeness is the dividing line.