Alphabeta MathCompact Operators and Riesz Schauder Theory — Examples
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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
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- 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
- Orthonormal Bases, Parseval and Fourier Series
- Schauder Bases Approximation and Banach Space Pathologies
- 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
- 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
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
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
- 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
- Completeness, Completion, and Uniform Continuity
- 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
Example
Diagonal operator on ell p is compact iff diagonal tends to zero
Example
Continuous kernel integral operator is compact on c of an interval
Counterexample
Identity is compact iff the space is finite dimensional
Counterexample
A compact operator can have nondense range
Example
Fredholm alternative for an integral equation
Counterexample
Compactness is not preserved by strong operator limits
Remark
Approximation property controls finite rank density in compact operators
Sources
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 p.70, example after Theorem 3.2
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.186, Example 4.26
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1 p.71, Lemma 3.4
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2, integral operators with continuous kernels
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.184, the identity is compact only in finite dimension
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1, examples and counterexamples for compactness
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1, finite-rank examples of compact operators
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.184, Example 4.23
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §6.5 p.189, Theorem 6.30 and its integral-equation discussion
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.4 p.198, Remark 4.42
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2, strong limits of finite-rank projections
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §3.1, compactness is not preserved by strong limits
- Theo Bühler and Dietmar Salamon, Functional Analysis — §4.2 p.188, Exercise 4.29 forward direction