Topological Vector Spaces (Hardcover, 2)

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.