Locally Convex Spaces
著者
書誌事項
Locally Convex Spaces
(Mathematische Leitfäden)
Springer Fachmedien, 1981
- : pbk
大学図書館所蔵 全1件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
収録内容
- "Originally published by B.G. Teubner, Stuttgart in 1981"--T.p. verso
内容説明・目次
内容説明
The present book grew out of several courses which I have taught at the University of Zurich and at the University of Maryland during the past seven years. It is primarily intended to be a systematic text on locally convex spaces at the level of a student who has some familiarity with general topology and basic measure theory. However, since much of the material is of fairly recent origin and partly appears here for the first time in a book, and also since some well-known material has been given a not so well-known treatment, I hope that this book might prove useful even to more advanced readers. And in addition I hope that the selection ofmaterial marks a sufficient set-offfrom the treatments in e.g. N. Bourbaki [4], [5], R.E. Edwards [1], K. Floret-J. Wloka [1], H.G.Garnir-M.De Wilde-J.Schmets [1], AGrothendieck [13], H.Heuser [1], J. Horvath [1], J. L. Kelley-I. Namioka et al. [1], G. Kothe [7], [10], A P. Robertson- W.Robertson [1], W.Rudin [2], H.H.Schaefer [1], F.Treves [l],A Wilansky [1]. A few sentences should be said about the organization of the book. It consists of 21 chapters which are grouped into three parts. Each chapter splits into several sections.
Chapters, sections, and the statements therein are enumerated in consecutive fashion.
目次
I: Linear Topologies.- 1 Vector Spaces.- 1.1 Generalities.- 1.2 Elementary Constructions.- 1.3 Linear Maps.- 1.4 Linear Independence.- 1.5 Linear Forms.- 1.6 Bilinear Maps and Tensor Products.- 1.7 Some Examples.- 2 Topological Vector Spaces.- 2.1 Generalities.- 2.2 Circled and Absorbent Sets.- 2.3 Bounded Sets. Continuous Linear Forms.- 2.4 Projective Topologies.- 2.5 A Universal Characterization of Products.- 2.6 Projective Limits.- 2.7 F-Seminorms.- 2.8 Metrizable Tvs.- 2.9 Projective Representation of Tvs.- 2.10 Linear Topologies on Function and Sequence Spaces.- 2.11 References.- 3 Completeness.- 3.1 Some General Concepts.- 3.2 Some Completeness Concepts.- 3.3 Completion of a Tvs.- 3.4 Extension of Uniformly Continuous Maps.- 3.5 Precompact Sets.- 3.6 Examples.- 3.7 References.- 4 Inductive Linear Topologies.- 4.1 Generalities.- 4.2 Quotients of Tvs.- 4.3 Direct Sums.- 4.4 Some Completeness Results.- 4.5 Inductive Limits.- 4.6 Strict Inductive Limits.- 4.7 References.- 5 Baire Tvs and Webbed Tvs.- 5.1 Baire Category.- 5.2 Webs in Tvs.- 5.3 Stability Properties of Webbed Tvs.- 5.4 The Closed Graph Theorem.- 5.5 Some Consequences.- 5.6 Strictly Webbed Tvs.- 5.7 Some Examples.- 5.8 References.- 6 Locally r-Convex Tvs.- 6.1 r-Convex Sets.- 6.2 r-Convex Sets in Tvs.- 6.3 Gauge Functionals and r-Seminorms.- 6.4 Continuity Properties of Gauge Functionals.- 6.5 Definition and Basic Properties of Lc,s.- 6.6 Some Permanence Properties of Lc,s.- 6.7 Bounded, Precompact, and Compact Sets.- 6.8 Locally Bounded Tvs.- 6.9 Linear Mappings Between r-Normable Tvs.- 6.10 Examples.- 6.11 References.- 7 Theorems of Hahn-Banach, Krein-Milman, and Riesz.- 7.1 Sublinear Functionals.- 7.2 Extension Theorem for Lcs.- 7.3 Separation Theorems.- 7.4 Extension Theorems for Normed Spaces.- 7.5 The Krein-Milman Theorem.- 7.6 The Riesz Representation Theorem.- 7.7 References.- II: Duality Theory for Locally Convex Spaces.- 8 Basic Duality Theory.- 8.1 Dual Pairings and Weak Topologies.- 8.2 Polarization.- 8.3 Barrels and Disks.- 8.4 Bornologies and ?-Topologies.- 8.5 Equicontinuous Sets and Compactologies.- 8.6 Continuity of Linear Maps.- 8.7 Duality of Subspaces and Quotients.- 8.8 Duality of Products and Direct Sums.- 8.9 The Stone-Weierstrass Theorem.- 8.10 References.- 9 Continuous Convergence and Related Topologies.- 9.1 Continuous Convergence.- 9.2 Grothendieck's Completeness Theorem.- 9.3 The Topologies ?t and ?.- 9.4 The Banach-Dieudonne Theorem.- 9.5 B-Completeness and Related Properties.- 9.6 Open and Nearly Open Mappings.- 9.7 Application to B-Completeness.- 9.8 On Weak Compactness.- 9.9 References.- 10 Local Convergence and Schwartz Spaces.- 10.1 ?-Convergence. Local Convergence.- 10.2 Local Completeness.- 10.3 Equicontinuous Convergence. The Topologies ?t and ?.- 10.4 Schwartz Topologies.- 10.5 A Universal Schwartz Space.- 10.6 Diametral Dimension. Power Series Spaces.- 10.7 Quasi-Normable Lcs.- 10.8 Application to Continuous Function Spaces.- 10.9 References.- 11 Barrelledness and Reflexivity.- 11.1 Barrelled Lcs.- 11.2 Quasi-Barrelled Lcs.- 11.3 Some Permanence Properties.- 11.4 Semi-Reflexive and Reflexive Lcs.- 11.5 Semi-Montel and Montei Spaces.- 11.6 On Frechet-Montel Spaces.- 11.7 Application to Continuous Function Spaces.- 11.8 On Uniformly Convex Banach Spaces.- 11.9 On Hilbert Spaces.- 11.10 References.- 12 Sequential Barrelledness.- 12.1 ??-Barrelled and c0-Barrelled Lcs.- 12.2 ?0-Barrelled Lcs.- 12.3 Absorbent and Bornivorous Sequences.- 12.4 DF-Spaces, gDF-Spaces, and df-Spaces.- 12.5 Relations to Schwartz Topologies.- 12.6 Application to Continuous Function Spaces.- 12.7 References.- 13 Bornological and Ultrabornological Spaces.- 13.1 Generalities.- 13.2 ?-Convergent and Rapidly ?-Convergent Sequences.- 13.3 Associated Bornological and Ultrabornological Spaces.- 13.4 On the Topology ?(E', E)bor.- 13.5 Permanence Properties.- 13.6 Application to Continuous Function Spaces.- 13.7 References.- 14 On Topological Bases.- 14.1 Biorthogonal Sequences.- 14.2 Bases and Schauder Bases.- 14.3 Weak Bases. Equicontinuous Bases.- 14.4 Examples and Additional Remarks.- 14.5 Shrinking and Boundedly Complete Bases.- 14.6 On Summable Sequences.- 14.7 Unconditional and Absolute Bases.- 14.8 Orthonormal Bases in Hilbert Spaces.- 14.9 References.- III Tensor Products and Nuclearity.- 15 The Projective Tensor Product.- 15.1 Generalities on Projective Tensor Products.- 15.2 Tensor Product and Linear Mappings.- 15.3 Linear Mappings with Values in a Dual.- 15.4 Projective Limits and Projective Tensor Products.- 15.5 Inductive Limits and Projective Tensor Products.- 15.6 Some Stability Properties.- 15.7 Projective Tensor Products with ?1 (?)-spaces.- 15.8 References.- 16 The Injective Tensor Product.- 16.1 ?-Products and ?-Tensor Products.- 16.2 Tensor Product and Linear Mappings.- 16.3 Projective and Inductive Limits.- 16.4 Some Stability Properties.- 16.5 Spaces of Summable Sequences.- 16.6 Continuous Vector Valued Functions.- 16.7 Holomorphic Vector Valued Functions.- 16.8 References.- 17 Some Classes of Operators.- 17.1 Compact Operators.- 17.2 Weakly Compact Operators.- 17.3 Nuclear Operators.- 17.4 Integral Operators.- 17.5 The Trace for Finite Operators.- 17.6 Some Particular Cases.- 17.7 References.- 18 The Approximation Property.- 18.1 Generalities.- 18.2 Some Stability Properties.- 18.3 The Approximation Property for Banach Spaces.- 18.4 The Metric Approximation Property.- 18.5 The Approximation Property for Concrete Spaces.- 18.6 References.- 19 Ideals of Operators in Banach Spaces.- 19.1 Generalities.- 19.2 Dual, Injective, and Surjective Ideals.- 19.3 Ideal-Quasinorms.- 19.4 ?p-Sequences.- 19.5 Absolutely p-Summing Operators.- 19.6 Factorization.- 19.7 p-Nuclear Operators.- 19.8 p-Approximable Operators.- 19.9 Strongly Nuclear Operators.- 19.10 Some Multiplication Theorems.- 19.11 References.- 20 Components of Ideals on Particular Spaces.- 20.1 Compact Operators on Hilbert Spaces.- 20.2 The Schatten-von Neumann Classes.- 20.3 Grothendieck's Inequality.- 20.4 Applications.- 20.5$$ {P_p}and{N_q} $$on Hilbert Spaces.- 20.6 Composition of Absolutely Summing Operators.- 20.7 Weakly Compact Operators on T(K)-Spaces.- 20.8 References.- 21 Nuclear Locally Convex Spaces.- 21.1 Locally Convex A-Spaces.- 21.2 Generalities on Nuclear Spaces.- 21.3 Further Characterizations by Tensor Products.- 21.4 Nuclear Spaces and Choquet Simplexes.- 21.5 On Co-Nuclear Spaces.- 21.6 Examples of Nuclear Spaces.- 21.7 A Universal Generator.- 21.8 Strongly Nuclear Spaces.- 21.9 Associated Topologies.- 21.10 Bases in Nuclear Spaces.- 21.11 References.- List of Symbols.
「Nielsen BookData」 より