Alphabeta MathHilbert Space Geometry and Riesz Representation
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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 Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Dual Spaces Adjoint Operators and Annihilators
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
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
- 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
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- Weak Mixing and the Chacon Transformation
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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 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
2 · Logical flowchart
3 · Items
Definition
Real and complex inner-product spaces and their induced length
Theorem
Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
Corollary
The induced length is a norm
Theorem
The parallelogram law
Theorem
Jordan–von Neumann: a norm is induced by an inner product exactly when it satisfies the parallelogram law
Definition
Hilbert space
Lemma
The inner product is jointly continuous
Theorem
The norm completion of an inner-product space is a Hilbert space
Definition
Orthogonality and the orthogonal complement
Lemma
Pythagoras and finite orthogonal sums
Lemma
Orthogonal complements are closed
Lemma
A minimizing sequence in a convex set is Cauchy
Theorem
Projection onto a nonempty closed convex set
Theorem
Variational characterisation of the nearest point
Theorem
Orthogonal decomposition by a closed subspace
Definition
The Hilbert orthogonal projection onto a closed subspace
Lemma
Hilbert projections are linear, self-adjoint and contractive
Theorem
The double orthogonal complement of a subspace is its closure
Theorem
Riesz representation for Hilbert spaces
Corollary
Hilbert spaces are reflexive
Definition
The Hilbert-space adjoint of a bounded operator
Theorem
Hilbert-adjoint identities
Definition
Self-adjoint, positive, unitary and normal operators
Lemma
Kernel–range orthogonality for Hilbert adjoints
Remark
Agreement with the concrete L-two projection
Remark
Lax–Milgram is owned by the PDE track
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, §2.3.6 and §§5.3.1–5.3.2
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lectures 15–16 and 22–23
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, Lemma 1.40, p.38
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lectures 15–16
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, pp.38–39
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, p.39
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 15
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 1.41, p.39
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 16
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Definitions 1.0.1 and 2.0.1
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, p.38
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 1.44, pp.40–41
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Theorem 2.0.4
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.37, p.238
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 180
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Corollary 2.0.5
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.36, p.237
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Definition 182 and Proposition 183
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.36 and Lemma 5.38, pp.237–238
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Proposition 183
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 181
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorems 1.43 and 5.35, pp.39 and 236
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 184
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, Theorem 2.0.6
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 5.35, p.236
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 185
- Theo Bühler and Dietmar Salamon, Functional Analysis, Definition 5.39, p.239
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 23
- Theo Bühler and Dietmar Salamon, Functional Analysis, Lemma 5.38, p.238