Chapter 02
Distributions as Functionals
Distributions defined as continuous linear functionals on the test space, the order of a distribution, and the duality pairing that replaces pointwise values.
Built · Not yet releasedCore analysis · Graduate course · Staged public deployment
Distribution Theory
Objects too singular to be functions, made rigorous.
Distribution theory extends calculus to objects too singular to be ordinary functions. Built on spaces of smooth test functions, it defines generalized functions as continuous linear functionals, gives every distribution derivatives of all orders, and makes the Dirac delta, the principal value, and weak solutions of differential equations precise. These six chapters develop the theory from test functions and the duality pairing through differentiation, support, and convergence.
6Chapters
24Visual investigations
180Exercises with corrections
GraduateLevel
Chapter 1 is free now.
Chapters 2 through 6 are built and will be released progressively as public deployment continues.
Where this course sits
Real Analysis→General Topology→Measure Theory→Distribution Theory→Sobolev Spaces & PDEs
What you will master
Three capabilities the course builds
I
Read singular objects rigorously
Treat the Dirac delta, the principal value, and the derivative of a jump as precise mathematical objects, not informal shorthand.
II
Differentiate without limits
Give every distribution derivatives of all orders, and compute them through integration by parts and jump formulas that classical calculus cannot reach.
III
Solve equations weakly
Use convergence in the distribution space and approximate identities to make weak solutions and fundamental solutions of differential equations precise.
The syllabus
Six chapters, built in sequence
All six chapters are built. Chapter 1 is released free now. Chapters 2 through 6 remain visible here as the course roadmap and will be released progressively. Each chapter includes visual investigations, complete definitions and proofs, thirty exercises, three progressive hints per exercise, and a detailed correction.
Chapter 01
Test Functions 𝒟(Ω)
Infinitely differentiable functions of compact support, the bump construction, mollifiers, and the precise notion of convergence that makes the test space a home for duality.
Free · Open chapterChapter 03
Regular & Singular Distributions
Locally integrable functions embedded as distributions, the Dirac delta and the Cauchy principal value, and exactly what separates a function from a genuine distribution.
Built · Not yet releasedChapter 04
Distributional Differentiation
Weak derivatives defined by moving the derivative onto the test function, the derivative of the Heaviside step, jump formulas, and derivatives of all orders.
Built · Not yet releasedChapter 05
Support, Order & Localization
Support and singular support, compactly supported distributions, partitions of unity, and the structure theorem for distributions supported at a point.
Built · Not yet releasedChapter 06
Convergence of Distributions
Weak-star convergence, sequential completeness, approximate identities converging to the delta, and the Dirac comb, with a bridge to tempered distributions and Fourier.
Built · Not yet releasedVisual first
Four premium investigations per chapter, on light crisp backgrounds.
Complete proofs
Every theorem stated with all hypotheses and proved in full.
Guided practice
Thirty exercises per chapter with three hints and a full correction each.
Bilingual
Full English and French presentation sharing one mathematical corpus.