Alphabeta MathBanach Alaoglu Goldstine and Krein Milman — 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
- 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
- Dual Spaces Adjoint Operators and Annihilators
- Geometric Hahn Banach and Convex Separation
- Locally Convex Spaces and Continuous Separation
- Norming and Separation under Hahn–Banach
- The Analytic Hahn Banach Theorem
- Weak and Weak Star Topologies
Linear Algebra
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- 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
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- limsup, liminf, and Subsequential Limits
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- 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
- 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
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Example
Weak-star compactness of probability measures
Example
Extreme points of the probability measures are Dirac masses
Example
Extreme points of the ell-infinity unit ball
Counterexample
The c0 unit ball has no extreme points
Corollary
c0 is not isometrically a dual space
Counterexample
Weak-star compact does not imply weak-star sequentially compact
Remark