.

0 (hodnocen0 x )

BK

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

000238139

Preface ...i // 1. Basic Concepts in Banach Spaces ...1 - Banach spaces, subspaces, quotient spaces, Holder and Minkowski inequalities, classical spaces Co, ip, Lp[0,1], C[0,1], finite-dimensional subspaces, Hilbert spaces, Riesz’ lemma, separability, orthonormal bases in Hilbert spaces. In Exercises: Geometric and topological properties of Banach and Hilbert spaces, unconditional convergence. // 2. Hahn-Banach Theorem, Dual Space ...27 - The Hahn-Banach theorem, dual space, dual operators, elementary Riesz’ representation of duals to c0, ip, Lp[0,1] and C[0,1]. In Exercises: Examples on duality, convexity and separation, the Banach limit. // 3. Weak Topologies, More on the Structure of Banach Spaces ...41 - Weak topology, weak star topology, second dual, bounded sets, Banach-Steinhaus theorems, Alaoglu’s theorem, Goldstine’s theorem, reflexivity, extreme points, the Krein-Milman theorem, James-Simons-Godefroy theorems on James boundaries of sets, James’ characterization of weak compactness, the Eberlein-Smulyan and Krein-Smulyan theorems in separable spaces. In Exercises: Geometric and topological properties of the weak and weak star topologies, extremal structure of convex sets, more on James’ results on weak compactness, sliding hump technique, dual spaces, separation. // 4. Open Mapping Theorem, More on Classical Spaces ...67 - Banach’s open mapping and closed graph theorems, projections and complementability, Auerbach bases, Sobczyk’s theorem on complementability of Co in separable overspaces, Phillips’ theorem on non-complementability of Co in ioo, every separable Banach space is isometric to a subspace of C[0,1], every separable Banach space is isomorphic to a quotient of t\, Schur’s theorem on the coincidence of weak and norm convergence of sequences in . In Exercises: subspaces, complementability, hyperplanes, projections, isomorphisms, isometries, operators with closed range, bilinear forms. //

5. Differentiability of Norms and Duality ...91 - Gateaux and Frechet differentiability of norms, Smulyan’s dual test, Kadets’s renorming of spaces with separable dual by Frechet differentiable norms, the Asplund-Lindenstrauss result on Frechet differentiability on dense set of convex functions on spaces with separable dual, Mazur’s theorem, Lindenstrauss’ results on strongly exposed points of weakly compact sets. In Exercises: Differentiability of convex functions and extremal structure of convex sets in classical spaces, rotundity, Kadets-Klee property, Fenchel duality, Mazur’s intersection property, strong sub differ entiality. // 6. Compact Operators on Banach Spaces ...119 - Finite rank and compact operators, spectrum and resolvent, self-adjoint operators on Hilbert spaces, spectral radius, spectral theory of compact self-adjoint operators on Hilbert spaces, normal operators, the Fredholm alternative, Fredholm operators, examples of integral operators. In Exercises: Examples on spectra, completely continuous and Hilbert-Schmidt operators, Lomonosov’s theorem on invariant subspaces, some properties of /C ) and // 7. Fixed Point Property ...149 - The Markov-Kakutani theorem, Banach’s contraction principle, non-expansive mappings in Hilbert spaces, Schauder’s theorems, applications to differential equations. In Exercises: Examples of fixed points, Alspach’s fixed-point-free isometry on a weakly compact set, Werner’s proof of the Markov-Kakutani theorem. //

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-quet’s representation theorem in metrizable case, the Banach-Dieudonne theorem, the Eberlein-Śmulyan 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-Ptak 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, Pełczyński’s theorems on subspaces of £p spaces, unconditional Schauder bases, James’ theorem on containment of £\ and cq , Pitt’s theorem, James’ space J, Khintchine’s inequality, Rademacher functions in Lv. 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-Pelczyhski’s factorization, Rosenthal’s characterization of Eberlein compacta, weakly Lindelóf 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 (Vasak) spaces, the Namioka property, other types of compacta. // 11. Uniform Rotundity, Finite Representability ...263 - Uniform rotundity, the modulus of rotundity, uniform Frechet differentiability, duality, uniform convexity of Lp spaces for p £ (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 super reflexive 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 // Index

(OCoLC)36638438

cnb000188944