1300 187 187
Though the search for good selectors dates back to the early twentieth century, selectors play an increasingly important role in current research. This book is the first to assemble the scattered literature into a coherent and elegant presentation of what is known and proven about selectors--and what remains to be found.
The authors focus on selection theorems that are related to the axiom of choice, particularly selectors of small Borel or Baire classes. After examining some of the relevant work of Michael and Kuratowski & Ryll-Nardzewski and presenting background material, the text constructs selectors obtained as limits of functions that are constant on the sets of certain partitions of metric spaces. These include selection theorems for maximal monotone maps, for the subdifferential of a continuous convex function, and for some geometrically defined maps, namely attainment and nearest-point maps.
Assuming only a basic background in analysis and topology, this book is ideal for graduate students and researchers who wish to expand their general knowledge of selectors, as well as for those who seek the latest results.
| Preface | p. vii |
| Introduction | p. ix |
| Classical results | p. 1 |
| Michael's Continuous Selection Theorem | p. 1 |
| Results of Kuratowski and Ryll-Nardzewski | p. 8 |
| Remarks | p. 13 |
| Functions that are constant on the sets of a disjoint discretely o-decomposable family of F s -sets | p. 19 |
| Discretely o-Decomposable Partitions of a Metric Space | p. 19 |
| Functions of the First Borel and Baire Classes | p. 25 |
| When is a Function of the First Borel Class also of the First Baire Class? | p. 39 |
| Remarks | p. 42 |
| Selectors for upper semi-continuous functions with non-empty compact values | p. 43 |
| A General Theorem | p. 45 |
| Special Theorems | p. 53 |
| Minimal Upper Semi-continuous Set-valued Maps | p. 53 |
| Remarks | p. 57 |
| Selectors for compact sets | p. 65 |
| A Sp ial Theorem | p. 67 |
| A Ge ral Theorem | p. 69 |
| Remarks | p. 88 |
| Applications | p. 91 |
| Monotone Maps and Maximal Monotone Maps | p. 95 |
| Subdifferential Maps | p. 101 |
| Attainment Maps from X* to X | p. 106 |
| Attainment Maps from X to X* | p. 107 |
| Metric Projections or Nearest Point Maps | p. 108 |
| Some Selections into Families of Convex Sets | p. 110 |
| Example | p. 118 |
| Remarks | p. 122 |
| Selectors for upper semi-continuous set-valued maps with nonempty values that are otherwise arbitrary | p. 123 |
| Diagonal Lemmas | p. 124 |
| Selection Theorems | p. 127 |
| A Selection Theorem for Lower Semi-continuous Set-valued Maps | p. 138 |
| Example | p. 140 |
| Remarks | p. 144 |
| Further applications | p. 147 |
| p. 149 | |
| Duals of Asplund Spaces | p. 151 |
| A Partial Converse to Theorem 5.4 | p. 156 |
| Remarks | p. 159 |
| Bibliography | p. 161 |
| Index | p. 165 |
| Table of Contents provided by Publisher. All Rights Reserved. |
ISBN: 9780691096285
ISBN-10: 0691096287
Audience:
Tertiary; University or College
Format:
Hardcover
Language:
English
Number Of Pages: 184
Published: 22nd July 2002
Dimensions (cm): 24.486 x 16.358
x 1.956
Weight (kg): 0.481
9am - 4pm AEST, Monday - Friday
