Alphabeta MathLocally Convex Spaces and Continuous Separation
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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
Linear Algebra
Real Analysis
- 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
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Complex Exponential and Euler's Formula
Set Theory
2 · Logical flowchart
3 · Items
Definition
Topological vector spaces over the real and complex fields
Lemma
Translations, dilations and absorption in a topological vector space
Definition
Local convexity, convex and balanced sets, and the continuous dual
Lemma
Convex closures and hulls of finitely many compact convex sets
Lemma
Open and closed balanced convex zero-neighborhood refinements
Definition
Minkowski gauge for an open convex zero-neighborhood
Lemma
Continuity, sublinearity and strict sublevels of an open convex gauge
Theorem
Continuous separation when one convex set is open
Theorem
The continuous dual separates points in a Hausdorff locally convex space
Theorem
Uniform strict separation of compact and closed convex sets
Sources
- Gerald Teschl, Topics in Real and Functional Analysis (17 November 2017)
- Theo Bühler and Dietmar Salamon, Functional Analysis (8 June 2017)
- Harald Hanche-Olsen, Topological vector spaces, version 1.6 (bibliographic origin; complete local argument replaces unavailable backing)
- Gerald Teschl, Topics in Real and Functional Analysis, section 5.1