Alphabeta MathSpectral Measures and Borel Functional Calculus
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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
Complex Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- 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
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Dual Spaces Adjoint Operators and Annihilators
- Gelfand Theory and Commutative C Star Algebras
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
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
- Density Separability and Convolution in Lᵖ
- 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 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)
- Arc Length and Rectifiable Curves
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- 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
- Sine, Cosine, and the Definition of Pi
- 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 Fundamental Theorems of Calculus
- The Logarithm and General Powers
- 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
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- 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
Projection valued measure
Lemma
Weak and strong additivity of orthogonal projections
Lemma
Scalar and complex measures from a pvm
Definition
Integral of a simple function against a pvm
Lemma
Simple pvm integral is representation independent
Theorem
Bounded borel pvm integral
Theorem
Pvm integral is a star homomorphism
Lemma
Continuous functional calculus produces a regular PVM
Theorem
Spectral theorem for bounded normal operators pvm form
Definition
Borel functional calculus for a bounded normal operator
Theorem
Borel functional calculus for bounded normal operators
Corollary
Spectral projections and resolution of the identity
Theorem
Support and uniqueness of the spectral measure
Definition
Cyclic vector and cyclic normal operator
Theorem
Cyclic spectral representation
Lemma
Maximal orthogonal family of cyclic reducing subspaces
Theorem
Multiplication operator form of the bounded normal spectral theorem
Definition
Spectral multiplicity function in the separable case
Lemma
Unitary intertwiners preserve direct-integral fiber dimension
Theorem
Unitary equivalence classified by measure class and multiplicity
Theorem
Stone resolvent formula for spectral projections
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.72, printed pp.273–276
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Definition 5.1 and Remark 5.5, pp.15–17
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.72 and §5.6.1, printed pp.273–277
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Definition 5.1 and Lemma 5.3, pp.15–16
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.76, printed pp.276–277
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Definition 5.1 and Proposition 5.3, pp.15–18
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.6.2, printed pp.277–279
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Proposition 5.3 and Definition 5.1, pp.16–18
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.6.2 and Lemma 5.77, printed pp.277–281
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Proposition 5.3, pp.17–18
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.73, printed pp.279–283
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Lemma 5.3 and Proposition 5.3, pp.16–18
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.73, printed pp.279–285
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.76 and Theorem 5.74, printed pp.283–288
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Theorem 5.6, pp.17–20
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.74, printed pp.288–291
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Corollary 5.7, pp.19–20
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.75, printed pp.291–293
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.75, printed pp.291–295
- Dana P. Williams, Lecture Notes on the Spectral Theorem, Theorem 5.6(c), p.18
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.81, printed pp.293–296
- Gerald Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., §4.1 and §6.3, printed pp.113–115 and 173–177
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.74 and §5.7, printed pp.288–296
- Andreas Kriegl, Funktionalanalysis, §8.61, printed pp.196–198
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.7, printed pp.290–296
- John B. Conway, A Course in Functional Analysis, 2nd ed., Chapter IX §10, printed pp.293–301
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.84, printed pp.294–296
- John B. Conway, A Course in Functional Analysis, 2nd ed., Chapter IX §10, printed pp.293–299
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.82 and §5.7, printed pp.293–296
- John B. Conway, A Course in Functional Analysis, 2nd ed., Chapter IX §10, printed pp.293–297
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.7 and Theorem 5.84, printed pp.293–296
- John B. Conway, A Course in Functional Analysis, 2nd ed., Chapter IX §10, printed pp.295–301
- Andreas Kriegl, Funktionalanalysis, §8.61 and §8.64, printed pp.196–199
- John B. Conway, A Course in Functional Analysis, 2nd ed., Propositions 10.17–10.19 and Theorem 10.21, printed pp.299–301
- Andreas Kriegl, Funktionalanalysis, §8.62–8.66, printed pp.197–200
- John B. Conway, A Course in Functional Analysis, 2nd ed., Theorems 10.20 and 10.21, printed pp.299–301
- Andreas Kriegl, Funktionalanalysis, Theorem 8.64 and Proposition 8.66, printed pp.198–200
- Gerald Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., Theorem 4.3, printed pp.113–115
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.79, printed pp.285–288