$26.00

# Topological Vector Spaces

## Preview

## Book Description

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.

## Table of Contents

**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.*

## Author(s)

### Biography

**Lawrence Narici** and** Edward Beckenstein** are professors of mathematics at St. John’s University in New York.

## Reviews

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 MATH1219

Praise 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