#functional-analysis

05 June 2026 · 5 min
The L-p Spaces
The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.
29 May 2026 · 13 min
The Karhunen-Loeve Expansion
The Karhunen-Loeve expansion writes a stochastic process in the eigenbasis of its covariance operator, coordinates that are uncorrelated and mean-square optimal. We prove the expansion and its optimality, then derive it for Brownian motion and the Brownian bridge.
17 May 2026 · 9 min
Mercer's Theorem and Reproducing Kernels
The spectral theorem made explicit for kernels. A continuous positive kernel gives a compact positive integral operator with continuous eigenfunctions, and Mercer's theorem expands it as a uniformly convergent eigenfunction series that builds the reproducing kernel Hilbert space.
16 May 2026 · 7 min
Compact Operators and the Spectral Theorem
The infinite-dimensional analogue of a symmetric matrix. Compact operators as norm limits of finite-rank ones, attainment of the norm at an eigenvector, and the spectral theorem diagonalising a compact self-adjoint operator by eigenvectors with eigenvalues tending to zero.
15 May 2026 · 6 min
Bounded Operators and the Adjoint
The algebra of operators on a Hilbert space. The operator norm and the completeness of the bounded operators, the adjoint built from the Riesz representation, the C-star identity, and the self-adjoint operators whose norm the quadratic form attains.
14 May 2026 · 6 min
Orthonormal Bases
When an orthonormal set spans a Hilbert space. The convergence of orthogonal series, the equivalent conditions for an orthonormal basis with Parseval's identity, the Gram-Schmidt construction, and the isometry of every separable Hilbert space with the sequence space l-squared.
12 May 2026 · 5 min
Projection and Riesz Representation
The two theorems that make a Hilbert space usable. The projection theorem that a closed convex set has a unique nearest point, the orthogonal decomposition into a subspace and its complement, and the Riesz representation of every bounded linear functional.
11 May 2026 · 5 min
L-squared and Completeness
The two model infinite-dimensional Hilbert spaces, the square-integrable functions and the square-summable sequences, and the Riesz-Fischer theorem that makes them complete.
10 May 2026 · 6 min
Inner Product Spaces
Geometry on a vector space. Inner products, the Cauchy-Schwarz inequality, the norm they induce, the parallelogram law that characterises inner-product norms, orthogonality with Pythagoras and Bessel's inequality, and the definition of a Hilbert space.
08 May 2026 · 8 min
Metric and Normed Spaces
Distance, abstracted. Metric and normed spaces, open and closed sets, completeness and the Banach fixed-point theorem, compactness and Heine-Borel, the equivalence of all norms in finite dimensions, and that a continuous function on a compact set attains its extremes.

05 June 2026 · 5 min

The L-p Spaces

The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.

29 May 2026 · 13 min

The Karhunen-Loeve Expansion

The Karhunen-Loeve expansion writes a stochastic process in the eigenbasis of its covariance operator, coordinates that are uncorrelated and mean-square optimal. We prove the expansion and its optimality, then derive it for Brownian motion and the Brownian bridge.

17 May 2026 · 9 min

Mercer's Theorem and Reproducing Kernels

The spectral theorem made explicit for kernels. A continuous positive kernel gives a compact positive integral operator with continuous eigenfunctions, and Mercer's theorem expands it as a uniformly convergent eigenfunction series that builds the reproducing kernel Hilbert space.

16 May 2026 · 7 min

Compact Operators and the Spectral Theorem

The infinite-dimensional analogue of a symmetric matrix. Compact operators as norm limits of finite-rank ones, attainment of the norm at an eigenvector, and the spectral theorem diagonalising a compact self-adjoint operator by eigenvectors with eigenvalues tending to zero.

15 May 2026 · 6 min

Bounded Operators and the Adjoint

The algebra of operators on a Hilbert space. The operator norm and the completeness of the bounded operators, the adjoint built from the Riesz representation, the C-star identity, and the self-adjoint operators whose norm the quadratic form attains.

14 May 2026 · 6 min

Orthonormal Bases

When an orthonormal set spans a Hilbert space. The convergence of orthogonal series, the equivalent conditions for an orthonormal basis with Parseval's identity, the Gram-Schmidt construction, and the isometry of every separable Hilbert space with the sequence space l-squared.

12 May 2026 · 5 min

Projection and Riesz Representation

The two theorems that make a Hilbert space usable. The projection theorem that a closed convex set has a unique nearest point, the orthogonal decomposition into a subspace and its complement, and the Riesz representation of every bounded linear functional.

11 May 2026 · 5 min

L-squared and Completeness

The two model infinite-dimensional Hilbert spaces, the square-integrable functions and the square-summable sequences, and the Riesz-Fischer theorem that makes them complete.

10 May 2026 · 6 min

Inner Product Spaces

Geometry on a vector space. Inner products, the Cauchy-Schwarz inequality, the norm they induce, the parallelogram law that characterises inner-product norms, orthogonality with Pythagoras and Bessel's inequality, and the definition of a Hilbert space.

08 May 2026 · 8 min

Metric and Normed Spaces

Distance, abstracted. Metric and normed spaces, open and closed sets, completeness and the Banach fixed-point theorem, compactness and Heine-Borel, the equivalence of all norms in finite dimensions, and that a continuous function on a compact set attains its extremes.