Chapter 02
Bounded Linear Operators
Continuity equals boundedness, the operator norm, the space B(X,Y) as a Banach space, invertibility, and the Neumann series for the resolvent.
Built · Not yet releasedCore analysis · Graduate course · Staged public deployment
Functional Analysis
Linear algebra and analysis, reunited in infinite dimensions.
Functional analysis studies vector spaces of functions and the linear operators between them, where completeness replaces finite dimension as the load-bearing hypothesis. Building on the topology and measure courses, it develops normed and Banach spaces, the four cornerstone theorems (Hahn-Banach, open mapping, closed graph, uniform boundedness), duality and weak topologies, the geometry of Hilbert space, compact operators and the Fredholm alternative, and closes with the spectral theorem - the infinite-dimensional analogue of diagonalisation and the language of quantum mechanics and partial differential equations.
10Chapters
GraduateLevel
EN · FRBilingual
Chapter 1 is free now.
All 10 chapters are built. Chapters 2 through 10 will be released progressively as public deployment continues.
What the course builds
Three capabilities the course builds
Build the machinery
Master the four cornerstone theorems - Hahn-Banach, open mapping, closed graph, and uniform boundedness - that turn completeness into working tools.
Think in infinite dimensions
See where finite-dimensional intuition survives and where it breaks: weak topologies, non-compact unit balls, and duals that are strictly larger.
Reach the spectral theorem
Carry Hilbert-space geometry through compact operators and the Fredholm alternative to eigenfunction expansions and the spectral theorem.
Before you start
Prerequisites
Functional analysis rests on the two foundational courses. Topology supplies open sets, continuity, and compactness in the abstract; measure theory supplies the L^p spaces and the integration that makes the function spaces complete. Linear algebra over the reals and complexes is assumed.
The syllabus
Ten chapters, built in sequence
All 10 chapters are built. Chapter 1 is released free now. Chapters 2 through 10 remain visible here as the course roadmap and will be released progressively. Each chapter includes visual investigations, complete definitions and proofs, graded exercises, progressive hints, and detailed corrections.
Chapter 01
Normed and Banach Spaces
Norms and the geometry they impose, the sequence and function spaces l^p, c_0 and C[a,b], equivalence of norms in finite dimensions, and completeness as the working hypothesis of analysis.
Free · Open chapterChapter 03
The Hahn-Banach Theorem
Extension of linear functionals dominated by a sublinear functional, the geometric separation of convex sets, and the resulting richness of the dual space.
Built · Not yet releasedChapter 04
Baire Category and the Three Theorems
The Baire category theorem and the two structural pillars it powers: the open mapping theorem and the closed graph theorem for maps between Banach spaces.
Built · Not yet releasedChapter 05
The Uniform Boundedness Principle
Banach-Steinhaus, the passage from pointwise to uniform bounds, and its consequences for the convergence of families of operators.
Built · Not yet releasedChapter 06
Dual Spaces and Weak Topologies
The duals of l^p and L^p, weak and weak-star convergence, reflexivity, and the Banach-Alaoglu theorem on compactness of the dual ball.
Built · Not yet releasedChapter 07
Hilbert Spaces
Inner products and orthogonality, projection onto closed convex sets, orthonormal bases, Bessel and Parseval, and the completeness that makes geometry work.
Built · Not yet releasedChapter 08
The Riesz Representation Theorem
The self-duality of Hilbert space through Riesz-Frechet, the construction of the adjoint operator, and self-adjoint, unitary, and normal operators.
Built · Not yet releasedChapter 09
Compact Operators
Compact operators and approximation by finite-rank operators, the Fredholm alternative, and integral operators as the motivating example.
Built · Not yet releasedChapter 10
The Spectral Theorem
Spectrum and resolvent, the spectral theorem for compact self-adjoint operators, eigenfunction expansions, and the bridge to the theory of self-adjoint operators.
Built · Not yet released