.

0 (hodnocen0 x )

BK

Singapore : World Scientific, 1998

x, 559 s. ; 23 cm

ISBN 981-02-3180-6 (váz.)

Series in pure mathematics ; vol. 24

Obsahuje bibliografické odkazy

000188253

VII // TABLE OF CONTENTS // PREFACE...v // 1. PRINCIPLES OF SET THEORY AND MATHEMATICAL // LOGIC...1 // 1.1. Basic Definitions...1 // 1.2. Equinumerosity...13 // 1.3. Predicate Calculus...22 // 1.4. Axiomatic Model Theory...29 // 1.5. Common Logic and Non-Monotone Logic...34 // 1.6. Axiomatic Set Theory...35 // 1.7. Higher Order Predicate Calculus...42 // 1.8. Cardinal Numbers...44 // 1.9. Axiomatic Number Theory...46 // 1.10. Well Ordered Spaces...55 // 1.11. Ordinal Numbers...59 // 1.12. Real Numbers...67 // 1.13. Complex Numbers...76 // 1.14. Category Theory...80 // 1.15. Philosophical Implications...83 // References...102 // 2. METRIC AND TOPOLOGICAL SPACES...108 // 2.1. Metric Spaces...108 // 2.2. Topological Spaces...121 // 2.3. Topology and Continuous Mappings of Metric Spaces.131 // 2.4. On Topological Groups... 138 // 2.5. Operations over Sets in Topological Spaces...142 // 2.6. Operations over Sets in Metric Spaces...146 // 2.7. Complete Metric Spaces...149 // 2.8. Products of Topological Spaces...154 // viii // 2.9. Connectedness...156 // 2.10. Compactness...166 // 2.11. Approximation...175 // 2.12. Metrizability...178 // 2.13. Baire’s Category Theory...179 // 2.14. On Brouwer’s Contribution to Dimension Theory...180 // 2.15. On the Topology of Surfaces...182 // References...186 // 3. HOMOTOPY THEORY...189 // 3.1. The Space of Continuous Mappings...189 // 3.2. Homotopy...190 // 3.3. Extension of Mappings...198 // 3.4. Retraction...199 // 3.5. Homotopy Groups...201

// 3.6. The Fundamental Group...203 // 3.7. Higher Homotopy Groups...204 // References...207 // 4. MEASURE SPACES...209 // 4.1. Measures...209 // 4.2. Integrals...211 // 4.3. Topological Measures...214 // 4.4. The Lebesgue Measure...215 // References...216 // 5. LINEAR SPACES...218 // 5.1. Fundamentals...218 // 5.2. Normed Spaces...221 // 5.3. Inner Product Spaces...231 // 5.4. Convexity...239 // 5.5. Characterizations of Inner Product Spaces...251 // 5.6. Geometry of the Sphere...256 // 5.7. Spectral Theory...261 // 5.8. Topological Vector Spaces...270 // 5.9. Smooth Mappings of Banach Spaces...286 // 5.10. Distributions and Fourier Analysis...294 // 5.11. Stability of Mappings and Functional Equations...306 // 5.12. On Distance Preserving Transformations of Euclidean // Spaces and Isometries...332 // References...335 // 6. MANIFOLDS AND FIBER BUNDLES...347 // 6.1. The Concept of a Manifold...347 // 6.2. Differential Manifolds...348 // 6.3. Orientable Manifolds...353 // 6.4. Vector and Tensor Fields on Manifolds...358 // 6.5. Submanifolds...360 // 6.6. Lie Algebras and Lie Groups...362 // 6.7. Differential Forms: Pfaffians...366 // 6.8. Fiber Bundles...370 // 6.9. Differential Geometry...376 // 6.10. Manifolds of Mappings...388 // 6.11. Transversality of Mappings and Cobordism...393 // 6.12. Stable Structures on Manifolds...397 // References...401 // 7. TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS... 406 // 7.1. Degree Theory...406 // 7.2. Fixed Point Theory...416 // 7.3. Differentiable

Dynamics...419 // 7.4. Morse Theory...428 // 7.5. Lusternik-Schnirelman Theory...455 // 7.6. Atiyah-Singer Index Theory...458 // 7.7. Catastrophe Theory of R. Thom...461 // References...464 // 8. VARIATIONAL ANALYSIS IN THE LARGE...469 // 8.1. The Main Theme...469 // 8.2. The Dirichlet Problem and its Ramifications...470 // 8.3. Generalized Extremum Problems...479 // 8.4. Various Kinds of Non-Degeneracy and Riemannian // Metrics...483 // 8.5. Almost-Riemannian Structures on Banach Manifolds...488 // 8.6. Bifurcation Theory for One Parameter...492 // 8.7. Minimal Surfaces...494 // 8.8. Morse-Rassias Theory and Plateau’s Problem...523 // References...532 // 9. MATHEMATICAL MODELLING OF THE PHYSICAL // SPACE-TIME...536 // 9.1. Introduction: Minkowski World...536 // 9.2. Minkowski Model...538 // 9.3. Grassmannian and Generalized Gauss Mapping...540 // 9.4. Curved World...543 // 9.5. Spinors and Twistors...545 // 9.6. The Problem of Grand Unification and Superunification.548 // 9.7. The Problem of a Local Extension of Quantum // Formalism...553 // References...556