Alphabeta MathNorming and Separation under Hahn–Banach
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1 · Prerequisites
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness in Metric Spaces
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simple Field Extensions and the Construction of the Complex Numbers
- Suprema and Infima
- The Analytic Hahn Banach Theorem
- The ZFC Axioms and the Basic Set Constructions
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
The real dominated-extension principle as an additional hypothesis over ZF
Theorem
Dominated extension conditional on the relative principle
Theorem
Relative norm-preserving Hahn–Banach extension over the real and complex fields
Corollary
Relative dual norming, point separation, and recovery of the norm
Lemma
Evaluation defines a bounded scalar-linear map into the bidual
Corollary
Relative Hahn–Banach makes the canonical bidual map an isometry
Definition
Convex sets and continuous real-hyperplane separation in a normed space
Definition
The finite gauge of an open convex neighbourhood of zero
Lemma
The open convex gauge is sublinear and recovers its set
Lemma
Relative separation of an open convex set from an exterior point
Lemma
A compact set and a disjoint closed set have a positive norm-distance gap
Theorem