SYSTEMATIC MATHEMATICS

Functional Analysis

This course contains 32 visible knowledge chapters in six content-sized units of lengths 7, 4, 6, 4, 5, and 6. What changes when a vector has infinitely many coordinates, or when one “vector” is an entire function? Begin by reading ℓ¹, ℓ², ℓ∞, C[0,1], and Lᵖ as concrete spaces of sequences or functions, calculating two norms by hand, and seeing why the inclusion rule for sequence spaces cannot be copied blindly to function spaces. Then ask how size is measured, whether approximations remain inside the space, and how a linear rule changes size. Normed, Banach, and Hilbert spaces are precise answers, not vocabulary to memorize first. The course builds complete operator spaces, dual and bidual measurements, weak convergence and Mazur averages, projections, bounded self-adjoint spectra, compact operators, weak PDEs, Galerkin error control, and domain-sensitive unbounded adjoints one definition at a time. Every theorem states its space, topology, and operator domain, and every finite-dimensional intuition meets an explicit infinite-dimensional failure.

Before this course: Completed Linear Algebra, Real Analysis, Measure & Lebesgue Integration, Topology, Partial Differential Equations, Numerical Analysis, and Calculus of Variations. Learners should be comfortable with rigorous proof, metric and topological compactness, Lᵖ spaces, complex scalar arithmetic, eigenstructure, weak PDE formulations, and variational existence arguments. Weak, weak-star, bidual, and unbounded-adjoint concepts are rebuilt here before use.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Concrete infinite-dimensional vectors: sequences and functions

Objective: Give a sequence in ℓ² but not ℓ¹.

A vector need not be a short arrow or a finite column. If addition and scalar multiplication are defined component by component or point by point, an entire infinite sequence or function can be one vector; a norm then says what “small” and “close” mean.

First identify the objects: The symbol ℓᵖ (read “little ell p”) denotes sequences x=(x₁,x₂,…) with Σ|x_n|ᵖ<∞ for 1≤p<∞, using ||x||ₚ=(Σ|x_n|ᵖ)^{1/p}; ℓ∞ contains bounded sequences with ||x||∞=sup_n|x_n|. The space C[0,1] contains continuous functions on [0,1], while Lᵖ[0,1] contains measurable functions with finite p-power integral, identified when they differ only on a null set. The result to establish is: For sequences on the positive integers, if 1≤p<q<∞ then ℓᵖ⊆ℓ^q and ||x||_q≤||x||_p. Thus absolute summability implies square summability, but the converse can fail. Read each symbol as a statement about a named space, norm or topology, and convergence mode.

The worked question is: For x_n=2^{-n}, compute its ℓ¹ and ℓ² norms and explain what the two numbers measure. Begin with “The ℓ¹ norm adds absolute coordinates: Σ_{n=1}^∞2^{-n}=1.” and finish with “The first norm measures total absolute coordinate size; the second measures Euclidean energy spread across infinitely many coordinates.” Then compare the result with the nearby failure case before deciding what has actually been proved.

For sequences on the positive integers, if 1≤p<q<∞ then ℓᵖ⊆ℓ^q and ||x||_q≤||x||_p. Thus absolute summability implies square summability, but the converse can fail.

If x∈ℓᵖ, every coordinate satisfies |x_n|≤||x||ₚ; otherwise one coordinate alone would exceed the full p-power sum.

Normalize first to ||x||ₚ=1. Then |x_n|≤1, so |x_n|^q≤|x_n|ᵖ for q>p.

Sum over n to get ||x||_q^q≤1, then restore the original scale and obtain ||x||_q≤||x||ₚ.

For x_n=2^{-n}, compute its ℓ¹ and ℓ² norms and explain what the two numbers measure.

  1. The ℓ¹ norm adds absolute coordinates: Σ_{n=1}^∞2^{-n}=1.
  2. The squared ℓ² norm is Σ4^{-n}=(1/4)/(1−1/4)=1/3, so ||x||₂=1/√3.
  3. The first norm measures total absolute coordinate size; the second measures Euclidean energy spread across infinitely many coordinates.

Result: The same sequence is one vector in both spaces, with ||x||₁=1 and ||x||₂=1/√3. The norm changes the geometry even when the underlying coordinate list is unchanged.