Alphabeta MathBanach Valued Integration and the Radon Nikodym Property
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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
- 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
- Banach Alaoglu Goldstine and Krein Milman
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Geometric Hahn Banach and Convex Separation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Reflexivity and Eberlein Smulian
- Sequential Uniform Boundedness with Countable Choice
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Weak and Weak Star Topologies
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- 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
- 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
- Convex and Semicontinuous Functions on Rⁿ
- 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 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
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- 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
Banach-valued simple function and integral
Lemma
The Banach-valued simple integral is well defined
Definition
Strongly measurable Banach-valued function
Theorem
Pettis measurability criterion for strong measurability
Definition
Bochner-integrable function
Theorem
Bochner integrability criterion
Lemma
Bochner integral norm inequality
Theorem
Bochner dominated convergence theorem
Theorem
Bounded linear maps commute with Bochner integration
Definition
Banach-valued vector measure and variation
Lemma
Bounded variation of a vector measure is a finite measure
Lemma
A Bochner density defines an absolutely continuous vector measure
Definition
Radon--Nikodym property
Definition
Dentable bounded set and slice
Lemma
Dentable average ranges give vector-measure densities
Lemma
Nondentability produces a vector measure without density
Theorem
RNP--dentability characterization
Lemma
RNP is invariant under Banach-space isomorphism
Lemma
RNP is separably determined
Lemma
RNP may be tested on the Lebesgue interval
Lemma
Lipschitz curves and dominated interval vector measures
Lemma
AC supplies the countable and dependent choices used in Banach integration
Theorem
RNP and almost-everywhere differentiability of Lipschitz curves
Theorem
Separable dual spaces have the Radon--Nikodym property
Theorem
Hilbert spaces are reflexive by Riesz representation
Theorem
Reflexive spaces have the Radon--Nikodym property
Theorem
c₀ fails the Radon--Nikodym property
Theorem
L¹[0,1] fails the Radon--Nikodym property
Corollary
c₀ is not isomorphic to a dual space
Theorem
Dunford--Pettis for real L¹ on a finite measure space
Sources
- Gerald Teschl, Topics in Real and Functional Analysis
- Gilles Pisier, Martingales in Banach Spaces
- Jeff Cheeger and Bruce Kleiner, On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces
- Thomas J. Jech, The Axiom of Choice, §2.4
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations