Integral Representation Theory - Applications to Convexity, Banach Spaces and Potential Theory
von: Jaroslav Luke?, Jan Malý, Ivan Netuka, Jirí Spurný
Walter de Gruyter GmbH & Co.KG, 2010
ISBN: 9783110203219
Sprache: Englisch
732 Seiten, Download: 3452 KB
Format: PDF, auch als Online-Lesen
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Integral Representation Theory - Applications to Convexity, Banach Spaces and Potential Theory
| Introduction | 6 | ||
| Contents | 12 | ||
| Prologue | 18 | ||
| 1.1 The Korovkin theorem | 18 | ||
| 1.2 Notes and comments | 20 | ||
| Compact convex sets | 21 | ||
| 2.1 Geometry of convex sets | 22 | ||
| 2.2 Interlude: On the space M ( K ) | 39 | ||
| 2.3 Structures in convex sets | 43 | ||
| 2.4 Exercises | 57 | ||
| 2.5 Notes and comments | 66 | ||
| Choquet theory of function spaces | 69 | ||
| 3.1 Function spaces | 70 | ||
| 3.2 More about Korovkin theorems | 81 | ||
| 3.3 On the H - barycenter mapping | 83 | ||
| 3.4 The Choquet representation theorem | 84 | ||
| 3.5 In-between theorems | 87 | ||
| 3.6 Maximal measures | 90 | ||
| 3.7 Boundaries and the Simons lemma | 95 | ||
| 3.8 The Bishop–de Leeuw theorem | 98 | ||
| 3.9 Minimum principles | 101 | ||
| 3.10 Orderings and dilations | 103 | ||
| 3.11 Exercises | 112 | ||
| 3.12 Notes and comments | 122 | ||
| Affine functions on compact convex sets | 124 | ||
| 4.1 Affine functions and the barycentric formula | 124 | ||
| 4.2 Barycentric theorem and strongly affine functions | 130 | ||
| 4.3 State space and representation of affine functions | 137 | ||
| 4.4 Affine Baire-one functions on dual unit balls | 144 | ||
| 4.5 Exercises | 146 | ||
| 4.6 Notes and comments | 150 | ||
| Perfect classes of functions and representation of affine functions | 152 | ||
| 5.1 Generation of sets and functions | 153 | ||
| 5.2 Baire and Borel sets | 159 | ||
| 5.3 Baire and Borel mappings | 163 | ||
| 5.4 Perfect classes of functions | 166 | ||
| 5.5 Affinely perfect classes of functions | 167 | ||
| 5.6 Representation of H - affine functions | 171 | ||
| 5.7 Exercises | 176 | ||
| 5.8 Notes and comments | 183 | ||
| Simplicial function spaces | 185 | ||
| 6.1 Basic properties of simplicial spaces | 186 | ||
| 6.2 Characterizations of simplicial spaces | 193 | ||
| 6.3 Simplicial spaces as L1-preduals | 195 | ||
| 6.4 The weak Dirichlet problem and Ac(H)-exposed points | 197 | ||
| 6.5 The Dirichlet problem for a single function | 199 | ||
| 6.6 Special classes of simplicial spaces | 202 | ||
| 6.7 The Daugavet property of simplicial spaces | 213 | ||
| 6.8 Choquet simplices | 215 | ||
| 6.9 Restriction of function spaces | 221 | ||
| 6.10 Exercises | 222 | ||
| 6.11 Notes and comments | 230 | ||
| Choquet theory of function cones | 233 | ||
| 7.1 Function cones | 233 | ||
| 7.2 Maximal measures | 239 | ||
| 7.3 Representation theorem | 241 | ||
| 7.4 Simplicial cones | 244 | ||
| 7.5 Ordered compact convex sets and simplicial measures | 249 | ||
| 7.6 Exercises | 257 | ||
| 7.7 Notes and comments | 260 | ||
| Choquet-like sets | 261 | ||
| 8.1 Split and parallel faces | 261 | ||
| 8.2 H - extremal and H - convex sets | 263 | ||
| 8.3 Choquet sets, M -sets and P -sets | 267 | ||
| 8.4 H - exposed sets | 274 | ||
| 8.5 Weak topology on boundary measures | 276 | ||
| 8.6 Characterizations of simpliciality by Choquet sets | 279 | ||
| 8.7 Exercises | 285 | ||
| 8.8 Notes and comments | 290 | ||
| Topologies on boundaries | 291 | ||
| 9.1 Topologies generated by extremal sets | 291 | ||
| 9.2 Induced measures on Choquet boundaries | 295 | ||
| 9.3 Functions continuous in ext and max topologies | 301 | ||
| 9.4 Strongly universally measurable functions | 305 | ||
| 9.5 Facial topology generated by M -sets | 313 | ||
| 9.6 Exercises | 320 | ||
| 9.7 Notes and comments | 325 | ||
| Deeper results on function spaces and compact convex sets | 327 | ||
| 10.1 Boundaries | 328 | ||
| 10.2 Isometries of spaces of affine continuous functions | 337 | ||
| 10.3 Baire measurability and boundedness of affine functions | 340 | ||
| 10.4 Embedding of `1 | 352 | ||
| 10.5 Metrizability of compact convex sets | 355 | ||
| 10.6 Continuous affine images | 368 | ||
| 10.7 Several topological results on Choquet boundaries | 375 | ||
| 10.8 Convex Baire-one functions | 382 | ||
| 10.9 Function spaces with continuous envelopes | 387 | ||
| 10.10 Exercises | 395 | ||
| 10.11 Notes and comments | 401 | ||
| Continuous and measurable selectors | 406 | ||
| 11.1 The Lazar selection theorem | 406 | ||
| 11.2 Applications of the Lazar selection theorem | 411 | ||
| 11.3 The weak Dirichlet problem for Baire functions | 415 | ||
| 11.4 Pointwise approximation of maximal measures | 417 | ||
| 11.5 Measurable selectors | 419 | ||
| 11.6 Exercises | 429 | ||
| 11.7 Notes and comments | 433 | ||
| Constructions of function spaces | 436 | ||
| 12.1 Products of function spaces | 437 | ||
| 12.2 Inverse limits of function spaces | 457 | ||
| 12.3 Several examples | 472 | ||
| 12.4 Exercises | 494 | ||
| 12.5 Notes and comments | 503 | ||
| Function spaces in potential theory and the Dirichlet problem | 506 | ||
| 13.1 Balayage and the Dirichlet problem | 508 | ||
| 13.2 Boundary behavior of solutions | 513 | ||
| 13.3 Function spaces and cones in potential theory | 521 | ||
| 13.4 Dirichlet problem: solution methods | 534 | ||
| 13.5 Generalized Dirichlet problem and uniqueness questions | 554 | ||
| 13.6 Exercises | 563 | ||
| 13.7 Notes and comments | 572 | ||
| Applications | 580 | ||
| 14.1 Representation of convex functions | 581 | ||
| 14.2 Representation of concave functions | 584 | ||
| 14.3 Doubly stochastic matrices | 589 | ||
| 14.4 The Riesz–Herglotz theorem | 590 | ||
| 14.5 Typically real holomorphic functions | 592 | ||
| 14.6 Holomorphic functions with positive real part | 597 | ||
| 14.7 Completely monotonic functions | 603 | ||
| 14.8 Positive definite functions on discrete groups | 606 | ||
| 14.9 Range of vector measures | 610 | ||
| 14.10 The Stone–Weierstrass approximation theorem | 612 | ||
| 14.11 Invariant and ergodic measures | 614 | ||
| 14.12 Exercises | 620 | ||
| 14.13 Notes and comments | 622 | ||
| Appendix | 625 | ||
| A.1 Functional analysis | 625 | ||
| A.2 Topology | 632 | ||
| A.3 Measure theory | 641 | ||
| A.4 Descriptive set theory | 654 | ||
| A.5 Resolvable sets and Baire-one functions | 657 | ||
| A.6 The Laplace equation | 662 | ||
| A.7 The heat equation | 666 | ||
| A.8 Axiomatic potential theory | 669 | ||
| Bibliography | 686 | ||
| List of symbols | 712 | ||
| Index | 720 |