Alphabeta MathWeak Derivatives and Sobolev Spaces
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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
Differential Geometry
Functional Analysis
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Distributions Test Functions and Differentiation
- Fourier Transform Convolution and Approximate Identities
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Weak and Weak Star Topologies
Linear Algebra
- Complexification, Realification and Real Structures
- 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
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- 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 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
- Areas of Elementary Plane Figures
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- 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 Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Inverse Function Theorem Completed
- 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
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Locally integrable functions as regular distributions
Definition
Weak derivative of a locally integrable function
Lemma
Weak differentiation ignores null-set changes
Lemma
Uniqueness of a weak derivative as an almost-everywhere class
Definition
Integer-order Sobolev spaces and their norms
Lemma
Classical derivatives agree with weak derivatives
Lemma
Linearity, locality, and commutation of weak derivatives
Remark
Weak derivatives are represented distributional derivatives
Definition
Absolute continuity on almost every coordinate line
Lemma
Integration by parts for dual-exponent Sobolev functions
Lemma
The Sobolev norm descends to equivalence classes
Lemma
Sobolev functions paste across an overlap
Lemma
Weak Leibniz rule with a smooth factor
Lemma
Weak derivatives persist under local Lp limits
Theorem
Zero weak gradient gives componentwise constants
Corollary
Weak differentiation has a closed graph on its natural domains
Definition
The notation Hᵏ and the reserved zero-boundary symbol
Lemma
ACL representatives recover their weak gradients by Fubini
Lemma
Bounded restriction and cutoff localisation in Sobolev spaces
Lemma
Weak lower semicontinuity of the Sobolev norm
Theorem
Integer-order Sobolev spaces are Banach
Theorem
The ACL characterisation of W^1,p
Theorem
Hᵏ is a Hilbert space under the derivative-sum inner product
Corollary
One-dimensional W^1,p functions have unique absolutely continuous representatives
Theorem
Chain rule for a C¹ function with bounded derivative
Theorem
Chain rule for globally Lipschitz scalar maps of Sobolev functions
Corollary
Positive, negative, and truncated Sobolev functions
Corollary
Sobolev maxima and minima form a lattice
Sources
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §1.1
- Semyon Dyatlov, Lecture notes for 18.155 (2022), regular distributions
- John K. Hunter, Notes on Partial Differential Equations (2014), §3.1
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §1.2
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 §3.1
- Juha Kinnunen, Sobolev Spaces (Aalto University, 2026), Lemma 1.14
- John K. Hunter, Notes on Partial Differential Equations (2014), Proposition 3.17
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011)
- John K. Hunter, Notes on Partial Differential Equations (2014)
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 2 §2.6
- Juha Kinnunen, Sobolev Spaces (Aalto University, 2026)
- Juha Kinnunen, Sobolev Spaces (2026)
- MATH 712 Real Analysis Solutions to Take-home Midterm, Spring 2016, Part B
- John K. Hunter, Notes on Partial Differential Equations (2014), Chapter 3 §3.5
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §§1.2–1.3
- Juha Kinnunen, Sobolev Spaces (2026), Chapters 1–2
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Chapter 9 §9.1
- Juha Kinnunen, Sobolev Spaces (2026), Chapters 1-2
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), Chapter 8 §8.2
- Joa Weber, Introduction to Sobolev Spaces (UNICAMP lecture notes)
- Ben Johnsrude and Mateusz Kwaśnicki, answers to "Preimage of a null set by a non-monotonic absolutely continuous function"
- Juha Kinnunen, Sobolev Spaces (2026), Chapter 2