Topological Vector Spaces

Topological Vector Spaces

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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 more

Product details

  • Hardback | 628 pages
  • 160.02 x 236.22 x 35.56mm | 1,020.58g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • Revised
  • 2nd Revised edition
  • 6 black & white illustrations, 3 black & white tables
  • 1584888660
  • 9781584888666
  • 1,867,502

Review quote

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 2012a This 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 Kothe 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. -Jose Bonet, Zentralblatt MATH 1219 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 Kothe, Rendiconti del Circolo Matematico di Palermo, Series II, Volume 35, Number 3, September 1986show more

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 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 more

About Zuhair Nashed

Lawrence Narici and Edward Beckenstein are professors of mathematics at St. John's University in New more