Alphabeta MathSchauder Bases Approximation and Banach Space Pathologies
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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
- Finite Dimensional Normed Spaces and Riesz Lemma
- Geometric Hahn Banach and Convex Separation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Reflexivity and Eberlein Smulian
- 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
- Radon Measures and the Riesz Markov Kakutani Theorem
- 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
- Equivalent Forms of Completeness
- 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
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
Schauder basis and coordinate functionals
Definition
Partial-sum projections and basis constant
Lemma
The Schauder coefficient space is Banach
Theorem
Coordinate functionals of a Schauder basis are bounded
Corollary
A Banach space with a Schauder basis is separable
Definition
Unconditional convergence of a Banach-space series
Definition
Unconditional and conditional Schauder bases
Theorem
Equivalent forms of unconditional convergence
Definition
Approximation property and bounded approximation property
Lemma
Uniformly bounded pointwise-convergent operators converge uniformly on compact sets
Theorem
A Schauder basis implies the bounded approximation property
Definition
Finitely additive charges and total variation on the power set of N
Lemma
Finite-range sequences are uniformly dense in ell-infinity
Definition
The finitely additive integral on ell-infinity
Lemma
The finitely additive integral is well-defined and isometric
Theorem
The dual of ell-infinity is ba
Theorem
Existence of a shift-invariant mean on bounded sequences
Corollary
The countably additive part of ba is ell-one
Definition
James space
Lemma
The James formula defines a norm
Theorem
James space is complete and separable
Lemma
Dual and bidual models for James space
Theorem
The canonical image of James space has codimension one
Theorem
James space is isometrically isomorphic to its bidual
Definition
Enflo finite-expansion and localized-trace system
Lemma
Enflo's quantitative localized-trace obstruction
Lemma
Enflo's Walsh-block estimates and symmetry average
Lemma
Enflo's Walsh-block assembly
Theorem
Reflexive approximation property implies metric approximation property
Theorem
A separable reflexive Banach space without the approximation property
Lemma
Finite-dimensional Auerbach bases
Lemma
The Dvoretzky--Rogers finite-block estimate
Theorem
Dvoretzky--Rogers theorem
Corollary
Universal agreement of absolute and unconditional convergence
Sources
- Thomas Schlumprecht, Course Notes in Functional Analysis, Math 655
- Michael Müger, Introduction to Functional Analysis
- Per Enflo, A counterexample to the approximation problem in Banach spaces
- Gerald Teschl, Topics in Real and Functional Analysis, Problem 4.20
- Theo Bühler and Dietmar Salamon, Functional Analysis
- Raymond A. Ryan, Introduction to Tensor Products of Banach Spaces
- A. Dvoretzky and C. A. Rogers, Absolute and Unconditional Convergence in Normed Linear Spaces