Covering all of the key areas in the field, this second edition includes revised information on the Choquet boundary, reflexivity and approximation theory, strict and uniform norm convexities, and so on.
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.
BackgroundTopology Valuation Theory Algebra Linear Functionals Hyperplanes Measure Theory Normed SpacesCommutative 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 CompletionTopological 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 SpacesExamplesLocally 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 SpacesBounded 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 TheoremHellyDualityPaired SpacesWeak TopologiesPolarsAlaogluPolar TopologiesEquicontinuityTopologies of PairsPermanence in DualityOrthogonalsAdjointsAdjoints and ContinuitySubspaces and QuotientsOpenness of Linear MapsLocal Convexity and HBEPKrein-Milman and Banach-Stone TheoremsMidpoints and SegmentsExtreme PointsFacesKrein-Milman TheoremsThe Choquet BoundaryThe Banach-Stone TheoremSeparating MapsNon-Archimedean TheoremsBanach-Stone VariationsVector-Valued Hahn-Banach TheoremsInjective SpacesMetric Extension PropertyIntersection PropertiesThe Center-Radius PropertyMetric Extension = CRPWeak Intersection PropertyRepresentation TheoremSummaryNotesBarreled 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 DisksInductive LimitsStrict Inductive Limits and LF-SpacesInductive Limits of LCSBornological SpacesBanach Disks Bornological SpacesClosed Graph TheoremsMaps with Closed GraphsClosed Linear MapsClosed Graph TheoremsOpen Mapping TheoremsApplicationsWebbed SpacesClosed Graph TheoremsLimits on the Domain SpaceOther Closed Graph TheoremsReflexivityReflexivity BasicsReflexive SpacesWeak-Star Closed SetsEberlein-Smulian TheoremReflexivity of Banach SpacesNorm-Attaining FunctionalsParticular DualsSchauder BasesApproximation PropertiesNorm Convexities and ApproximationStrict ConvexityUniform ConvexityBest ApproximationUniqueness of HB ExtensionsStone-Weierstrass TheoremBibliographyIndexExercises appear at the end of each chapter.