Alphabeta MathDistributions Test Functions and Differentiation
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Functional Analysis
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Test function space d of an open set
Definition
Fixed support test function frechet space
Definition
Test function topology
Definition
Distribution
Definition
Regular distribution from a locally integrable function
Definition
Convolution of a distribution with a test function
Lemma
Test function cutoffs and euclidean localization
Lemma
Fixed support test function spaces are complete
Lemma
Test function lf topology universal property
Theorem
Test function operations are continuous
Theorem
Local finite order characterization of distributions
Definition
Order of a distribution on a compact set
Definition
Distributional derivative
Definition
Multiplication of a distribution by a smooth function
Theorem
Leibniz rule for distributions
Definition
Dirac delta and its derivatives
Theorem
Distributions form a sheaf
Definition
Support of a distribution
Theorem
Compactly supported distributions have global finite order
Theorem
Distributions supported at one point
Lemma
Bounded test function sets have common compact support
Theorem
Sequential convergence in test function space
Definition
Weak and strong topologies on distributions
Theorem
Locally integrable functions embed in distributions
Lemma
Strong distribution convergence implies weak convergence
Lemma
Riemann–Lebesgue comparison for distribution test integrands
Theorem
Distributional differentiation is continuous and commutes
Lemma
Distribution pairing with smooth parameter families
Theorem
Convolution with a test function is smooth
Theorem
Uniform finite order bounds for pointwise bounded distributions
Definition
Pullback of a distribution by a diffeomorphism
Definition
Tensor product of distributions
Lemma
Compactly supported distributions extend to smooth functions
Theorem
Sequential convergence of smooth multipliers and distributions
Theorem
Extension by zero for distributions with ambient closed support
Theorem
Translation invariant test function operators are convolutions
Lemma
Complex l one functionals on finite measure spaces have bounded densities
Theorem
Local structure of distributions as derivatives of continuous functions
Lemma
Compact support continuous primitive representation
Theorem
Global locally finite structure of distributions
Lemma
Finite sums of product tests are dense on product open sets
Theorem
Tensor product distributions and iterated pairings
Definition
Convolution of distributions when one has compact support
Lemma
Convolution of distributions is well defined under the support hypothesis
Theorem
Mollifier approximation in distributions
Theorem
A distribution with zero derivatives on a connected open set is constant
Corollary
Smooth functions are weakly dense in distributions
Theorem
Associativity of distribution convolution under compact support
Theorem
Closed bounded test function sets are compact
Sources
- Semyon Dyatlov, Lecture notes for 18.155 (2022)
- Razvan Gelca, Functional Analysis
- J. Lebl, Basic Analysis, Riemann Integral in Several Variables; Darboux criterion combined with the elementary Lebesgue simple-integral construction
- Razvan Gelca, Functional Analysis; complete Chapter 7 reading recorded in batch coverage