With many new concrete examples and historical notes, Topological Vector Spaces, Second Edition provides one of the most thorough and up-to-date treatments of the Hahn–Banach theorem. This edition explores the theorem’s connection with the axiom of choice, discusses the uniqueness of Hahn–Banach extensions, and includes an entirely new chapter on vector-valued Hahn–Banach theorems. It also considers different approaches to the Banach–Stone theorem as well as variations of the theorem.
The book covers locally convex spaces; barreled, bornological, and webbed spaces; and reflexivity. It traces the development of various theorems from their earliest beginnings to present day, providing historical notes to place the results in context. The authors also chronicle the lives of key mathematicians, including Stefan Banach and Eduard Helly.
Suitable for both beginners and experienced researchers, this book contains an abundance of examples, exercises of varying levels of difficulty with many hints, and an extensive bibliography and index.
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 Café
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.
Biography
Lawrence Narici and Edward Beckenstein are professors of mathematics at St. John’s University in New York.
Besides a general renovation, the text has improved the topics related to the Hahn-Banach theorem … there is a whole new chapter on vector-valued Hahn-Banach theorems and an enlarged presentation of the Banach-Stone theorems. The text remains a nice expository book on the fundamentals of the theory of topological vector spaces.
—Luis Manuel Sanchez Ruiz, Mathematical Reviews, Issue 2012aThis is a nicely written, easy-to-read expository book of the classical theory of topological vector spaces. … The proofs are complete and very detailed. … The comprehensive exposition and the quantity and variety of exercises make the book really useful for beginners and make the material more easily accessible than the excellent classical monographs by Köthe or Schaefer. … this is a well-written book, with comprehensive proofs, many exercises and informative new sections of historical character, that presents in an accessible way the classical theory of locally convex topological vector spaces and that can be useful especially for beginners interested in this topic.
—José Bonet, Zentralblatt MATH 1219Praise for the First Edition:
This is a very carefully written introduction to topological vector spaces. But it is more. The enthusiasm of the authors for their subject, their untiring efforts to motivate and explain the ideas and proofs, and the abundance of well-chosen exercises make the book an initiation into a fascinating new world. The reader will feel that he does not get only one aspect of this field but that he really gets the whole picture.
—Gottfried Köthe, Rendiconti del Circolo Matematico di Palermo, Series II, Volume 35, Number 3, September 1986
