Topological Vector Spaces, Second Edition

Topological Vector Spaces, Second Edition

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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

  • Electronic book text | 628 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • New edition
  • 2nd New edition
  • 3 Tables, black and white; 6 Illustrations, black and white
  • 1584888679
  • 9781584888673

Table of contents

BackgroundTopology Valuation Theory Algebra Linear Functionals Hyperplanes Measure Theory Normed Spaces Commutative Topological GroupsElementary ConsiderationsSeparation and Compactness Bases at 0 for Group Topologies Subgroups and Products Quotients S-Topologies Metrizability CompletenessCompleteness Function Groups Total BoundednessCompactness and Total Boundedness Uniform Continuity Extension of Uniformly Continuous Maps Completion Topological Vector SpacesAbsorbent and Balanced Sets Convexity-AlgebraicBasic PropertiesConvexity-Topological Generating Vector Topologies A Non-Locally Convex SpaceProducts and QuotientsMetrizability and CompletionTopological Complements Finite-Dimensional and Locally Compact SpacesExamples Locally Convex Spaces and SeminormsSeminormsContinuity of SeminormsGaugesSublinear FunctionalsSeminorm TopologiesMetrizability of LCSContinuity of Linear MapsThe Compact-Open TopologyThe Point-Open TopologyEquicontinuity and Ascoli's Theorem Products, Quotients, and CompletionOrdered Vector Spaces Bounded SetsBounded SetsMetrizabilityStability of Bounded SetsContinuity Implies Local BoundednessWhen Locally Bounded Implies ContinuousLiouville's TheoremBornologies Hahn-Banach TheoremsWhat Is It?The Obvious SolutionDominated and Continuous ExtensionsConsequencesThe Mazur-Orlicz TheoremMinimal Sublinear FunctionalsGeometric Form Separation of Convex SetsOrigin of the TheoremFunctional Problem SolvedThe Axiom of ChoiceNotes on the Hahn-Banach TheoremHelly DualityPaired SpacesWeak TopologiesPolarsAlaogluPolar TopologiesEquicontinuityTopologies of PairsPermanence in DualityOrthogonalsAdjointsAdjoints and ContinuitySubspaces and QuotientsOpenness of Linear MapsLocal Convexity and HBEP Krein-Milman and Banach-Stone TheoremsMidpoints and SegmentsExtreme PointsFacesKrein-Milman TheoremsThe Choquet BoundaryThe Banach-Stone TheoremSeparating MapsNon-Archimedean TheoremsBanach-Stone Variations Vector-Valued Hahn-Banach TheoremsInjective SpacesMetric Extension PropertyIntersection PropertiesThe Center-Radius PropertyMetric Extension = CRPWeak Intersection PropertyRepresentation TheoremSummaryNotes Barreled SpacesThe Scottish CafeS-Topologies for L(X, Y)Barreled SpacesLower SemicontinuityRare SetsMeager, Nonmeager, and BaireThe Baire Category TheoremBaire TVSBanach-Steinhaus TheoremA Divergent Fourier SeriesInfrabarreled SpacesPermanence PropertiesIncreasing Sequence of Disks Inductive LimitsStrict Inductive Limits and LF-SpacesInductive Limits of LCS Bornological SpacesBanach Disks Bornological Spaces Closed Graph TheoremsMaps with Closed GraphsClosed Linear MapsClosed Graph TheoremsOpen Mapping TheoremsApplicationsWebbed SpacesClosed Graph TheoremsLimits on the Domain SpaceOther Closed Graph Theorems ReflexivityReflexivity BasicsReflexive SpacesWeak-Star Closed SetsEberlein-Smulian TheoremReflexivity of Banach SpacesNorm-Attaining FunctionalsParticular DualsSchauder BasesApproximation Properties Norm Convexities and ApproximationStrict ConvexityUniform ConvexityBest ApproximationUniqueness of HB ExtensionsStone-Weierstrass Theorem Bibliography Index Exercises appear at the end of each more

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

About Lawrence Narici

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