Alphabeta MathReflexivity and Eberlein Smulian
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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
- 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
- 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
- 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 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
- 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 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
Theorem
Reflexive iff unit ball weakly compact
Theorem
A Banach space is reflexive if and only if its dual is reflexive
Theorem
Closed subspaces of reflexive spaces are reflexive
Theorem
Quotients of reflexive spaces are reflexive
Lemma
Complex Lp duality from real Lp duality
Theorem
Reflexivity of Lp for one less p less infinity
Definition
Relative weak compactness and three sequential notions
Lemma
Eberlein–Šmulian separable reduction
Lemma
Eberlein–Šmulian metrization on the relevant dual ball
Lemma
Countable compactness closes in the bidual
Theorem
Eberlein–Šmulian theorem
Corollary
Reflexivity is equivalent to weak subsequential compactness of bounded sequences
Definition
Schur property
Theorem
Real and complex ell one have the Schur property
Corollary
Ell one is not reflexive
Definition
Uniformly convex Banach space
Lemma
Uniform convexity gives unique asymptotic centers
Theorem
Milman–Pettis theorem
Lemma
James nonreflexivity sequence separated from an annihilator
Lemma
James convex-block norm-attainment criterion
Theorem
James reflexivity theorem
Lemma
Quantitative Bishop–Phelps support functional construction
Theorem
Bishop phelps
Theorem
Separable dual implies separable primal
Corollary
Separable reflexive space has separable dual
Lemma
Clarkson inequalities in both exponent ranges
Corollary
Lᵖ is uniformly convex for 1<p<∞
Lemma
Real and complex c₀ are Banach
Sources
- Bühler–Salamon, Functional Analysis
- Gerald Teschl, Topics in Real and Functional Analysis
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
- Gerald B. Folland, Real Analysis, 2nd ed.
- John K. Hunter, Measure Theory
- Haase, The Functional Analysis of Quantum Information Theory
- Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
- Gerald Teschl, Topics in Real and Functional Analysis (2017)
- Harald Hanche-Olsen, Topological vector spaces
- Teck-Cheong Lim, On Asymptotic Centers and Fixed Points of Nonexpansive Mappings
- Robert E. Megginson, An Introduction to Banach Space Theory (1998)
- Philip D. Loewen and Xianfu Wang, A Generalized Variational Principle, Canadian Journal of Mathematics 53 (2001), 1174–1193
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Theorem 3.26
- Bühler–Salamon, Functional Analysis, Theorem 2.73(i)
- Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Corollary 3.27
- Bühler–Salamon, Functional Analysis, Theorem 2.73(ii)
- Kuriyama, Miyagi, Okada and Miyoshi, Elementary proof of Clarkson's inequalities and their generalization
- Ken Kuriyama, Mitsuhiro Miyagi, Mari Okada and Tetsuhiko Miyoshi, Elementary proof of Clarkson's inequalities and their generalization