Alphabeta MathHilbert Space Geometry and Riesz Representation — 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
- 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
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
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
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- 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
- 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 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
Example
The standard inner products make K n, ell two and quotient L two Hilbert spaces
Example
Projection onto a finite-dimensional subspace by a Gram matrix
Example
Projection onto the constants is the mean
Example
Distance to a closed subspace
Counterexample
An inner-product space need not be complete
Counterexample
A norm need not satisfy the parallelogram law
Counterexample
Nearest-point maps to convex sets need not be linear
Example
Adjoints of shifts, multiplication and integral operators
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis, Example 1.42, p.39 and §2.3.6
- 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, pp.38–41
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Definition 182 and Proposition 183
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3 and §5.3.1
- 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.39–41
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Theorem 178
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3 and §2.3.6
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 16
- Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, p.39 and §2.3.6
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lecture 15
- Theo Bühler and Dietmar Salamon, Functional Analysis, Theorem 1.44, pp.40–41
- Theo Bühler and Dietmar Salamon, Functional Analysis, §5.3.1 and Example 5.35 ff.
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Example 186