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Bibliografická citace

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Vydání první
Praha : Matfyzpress, 1996
2 svazky ; 30 cm

ISBN 80-85863-14-6 (brožováno)
Obsahuje bibliografii na straně 319-323 a rejstřík
I. 158 stran -- II. strana 160-329
Vydavatel: Matematicko-fyzikální fakulta University Karlovy
Table of Contents // Preface...  // 1. Basic Concepts in Banach // “Tlí cloPl"d— . -bspaces, Hübe« spaces, Riesf iemma, sepacabiii.y orthoiěorniaě bŕses ěn Hübert spaces. In Excises, Geo„e,,c and topesca, ptopert.es of Banach and Hilbert spaces, unconditional convergence.  // 2. Hahn-Banach Theorem, Dual Space... // the Banach limit.  // 3 Weak Topologies, More on the Structure of Banach Spaces...•••••• // dual spaces, separation.  // 4. Open Mapping Theorem, More on Classical Spaces...’ ’ ’ ‘ // Banach’s open mapping and closed graph theorems  Phillips’ // bach bases, Sobers «i« spKe is isometric to // ..»s, —,  // closed range, bilinear forms.  // 5. Differentiability of Norms and Duality...’ " . // subdifferentiality. // ...11? // 6 Compact Operators on Banach Spaces... // Finite rank and compact operators, spectrum and resolvent, self-adjoint operators on Hilbert // Lomonosov’s theorem on invariant subspaces, some properties of /C(f2) and B(h). // ...149 // 7. Fixed Point Property... . // The Markov-Kakutani theorem, Banach’s contraction principle, »o.-eXpan e mappm m // ???: ofS :zr -, we , // proof of the Markov-Kakutani theorem. // Table of Contents // 8. Locally Convex Spaces...159 // The notion of a locally convex space, metrizability, normability, finite dimensionality, separation, the bipolar theorem, the Mackey-Arens-Katetov theorem, the space of distributions, Cho-quets representation theorem in metrizable case, the
Banach-Dieudonné theorem, the Eberlein-Šmulyan theorem, Kaplansky’s theorem on countable tightness of the weak topology of a Banach space, the Banach-Stone theorem. In Exercises: Examples, the Banach-Stone theorem, Sobolev spaces, Grothendieck-Pták results. // 9. Schauder Bases...185 // The notion of a Schauder basis, shrinking and boundedly complete bases, James’ theorems on the characterization of reflexivity in terms of bases, Mazur’s basic sequence theorem, the Krein-Milman-Rutman stability result, Pelczynski’s theorems on subspaces of Łp spaces, unconditional Schauder bases, James’ theorem on containment of Łi and Co, Pitt’s theorem, James’ space J, Khintchine’s inequality, Rademacher functions in Lp. In Exercises: Examples of Hamel and Schauder bases, isomorphisms, strictly singular operators, more on the James space. // 10. Weakly Compact Sets and Spaces They Generate...217 // Weakly compactly generated spaces, the notion of an Eberlein compact, the notion of a Marku-shevich’s basis, Amir-Lindenstrauss’ projectional resolutions of the identity on weakly compactly generated spaces, Davis-Figiel-Johnson-Pelczynski’s factorization, Rosenthal’s characterization of Eberlein compacta, weakly Lindelöf spaces (Preiss-Talagrand), scattered compacta, the notion of a uniform Eberlein compact and renorming of C(K) by uniformly Gateaux differentiable norms (Argyros-Benyamini-Farmaki-Starbird-Troyanski), quasicomplements (Gurarii-Kadets), Polish or Cech
complete balls in their weak topology (Godefroy, Edgar, Wheeler), absolutely summing operators, Pietsch’s lemma and the Dvoretzky-Rogers theorem, the Dunford-Pettis property. In Exercises: Examples, the three space property, weakly countably determined (Vašák) spaces, the Namioka property, other types of compacta. // 11. Uniform Rotundity, Finite Representability...263 // Uniform rotundity, the modulus of rotundity, uniform Fréchet differentiability, duality, uniform convexity of Lp spaces for p Ł (l,oo), Kadets’s result on unconditional bases in uniformly rotund spaces, the notion of finite representability, the local reflexivity principle, superreflexive spaces, Enflo’s renorming of superreflexive spaces, the Gurarii-Gurarii-James theorem on Schauder bases in sup er reflexi ve spaces. In Exercises: Examples on uniform convexity and smoothness, URED norms, the type and cotype of Banach spaces, Stegall’s results on trees in dual spaces, the original definition of Tsirelson’s space. // 12. Application of Smoothness in Banach Spaces...289 // Kadets’s result that all infinite-dimensional separable reflexive Banach spaces are mutually homeomorphic, Aharoni’s result that every separable Banach space is Lipschitz equivalent to a subset of Co, the Heinrich-Mankiewicz method of linearization of Lipschitz maps, the smooth variational principle and the Bishop-Phelps theorem as its consequence, Lindenstrauss’ result on the density of norm attaining operators, the Bonic-Frampton
result on smooth approximation in separable spaces, sub differentiability. In Exercises: Examples on the smooth variational principle, rotundity, the Aharoni-Lindenstrauss space, the Gorelik principle, the Haar measure. // References...319 // Index...325

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