Topological Vector Spaces

Abstract

Background Topology Valuation Theory Algebra Linear Functionals Hyperplanes Measure Theory Normed Spaces Commutative Topological Groups Elementary Considerations Separation and Compactness Bases at 0 for Group Topologies Subgroups and Products Quotients S-Topologies Metrizability Completeness Completeness Function Groups Total Boundedness Compactness and Total Boundedness Uniform Continuity Extension of Uniformly Continuous Maps Completion Topological Vector Spaces Absorbent and Balanced Sets Convexity-Algebraic Basic Properties Convexity-Topological Generating Vector Topologies A Non-Locally Convex Space Products and Quotients Metrizability and Completion Topological Complements Finite-Dimensional and Locally Compact Spaces Examples Locally Convex Spaces and Seminorms Seminorms Continuity of Seminorms Gauges Sublinear Functionals Seminorm Topologies Metrizability of LCS Continuity of Linear Maps The Compact-Open Topology The Point-Open Topology Equicontinuity and Ascoli's Theorem Products, Quotients, and Completion Ordered Vector Spaces Bounded Sets Bounded Sets Metrizability Stability of Bounded Sets Continuity Implies Local Boundedness When Locally Bounded Implies Continuous Liouville's Theorem Bornologies Hahn-Banach Theorems What Is It? The Obvious Solution Dominated and Continuous Extensions Consequences The Mazur-Orlicz Theorem Minimal Sublinear Functionals Geometric Form Separation of Convex Sets Origin of the Theorem Functional Problem Solved The Axiom of Choice Notes on the Hahn-Banach Theorem Helly Duality Paired Spaces Weak Topologies Polars Alaoglu Polar Topologies Equicontinuity Topologies of Pairs Permanence in Duality Orthogonals Adjoints Adjoints and Continuity Subspaces and Quotients Openness of Linear Maps Local Convexity and HBEP Krein-Milman and Banach-Stone Theorems Midpoints and Segments Extreme Points Faces Krein-Milman Theorems The Choquet Boundary The Banach-Stone Theorem Separating Maps Non-Archimedean Theorems Banach-Stone Variations Vector-Valued Hahn-Banach Theorems Injective Spaces Metric Extension Property Intersection Properties The Center-Radius Property Metric Extension = CRP Weak Intersection Property Representation Theorem Summary Notes Barreled Spaces The Scottish Cafe S-Topologies for L(X, Y) Barreled Spaces Lower Semicontinuity Rare Sets Meager, Nonmeager, and Baire The Baire Category Theorem Baire TVS Banach-Steinhaus Theorem A Divergent Fourier Series Infrabarreled Spaces Permanence Properties Increasing Sequence of Disks Inductive Limits Strict Inductive Limits and LF-Spaces Inductive Limits of LCS Bornological Spaces Banach Disks Bornological Spaces Closed Graph Theorems Maps with Closed Graphs Closed Linear Maps Closed Graph Theorems Open Mapping Theorems Applications Webbed Spaces Closed Graph Theorems Limits on the Domain Space Other Closed Graph Theorems Reflexivity Reflexivity Basics Reflexive Spaces Weak-Star Closed Sets Eberlein-Smulian Theorem Reflexivity of Banach Spaces Norm-Attaining Functionals Particular Duals Schauder Bases Approximation Properties Norm Convexities and Approximation Strict Convexity Uniform Convexity Best Approximation Uniqueness of HB Extensions Stone-Weierstrass Theorem Bibliography Index Exercises appear at the end of each chapter.