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Selectors - John E. Jayne

Hardcover

Published: 1st July 2002
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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.

Prefacep. vii
Introductionp. ix
Classical resultsp. 1
Michael's Continuous Selection Theoremp. 1
Results of Kuratowski and Ryll-Nardzewskip. 8
Remarksp. 13
Functions that are constant on the sets of a disjoint discretely o-decomposable family of F s -setsp. 19
Discretely o-Decomposable Partitions of a Metric Spacep. 19
Functions of the First Borel and Baire Classesp. 25
When is a Function of the First Borel Class also of the First Baire Class?p. 39
Remarksp. 42
Selectors for upper semi-continuous functions with non-empty compact valuesp. 43
A General Theoremp. 45
Special Theoremsp. 53
Minimal Upper Semi-continuous Set-valued Mapsp. 53
Remarksp. 57
Selectors for compact setsp. 65
A Sp ial Theoremp. 67
A Ge ral Theoremp. 69
Remarksp. 88
Applicationsp. 91
Monotone Maps and Maximal Monotone Mapsp. 95
Subdifferential Mapsp. 101
Attainment Maps from X* to Xp. 106
Attainment Maps from X to X*p. 107
Metric Projections or Nearest Point Mapsp. 108
Some Selections into Families of Convex Setsp. 110
Examplep. 118
Remarksp. 122
Selectors for upper semi-continuous set-valued maps with nonempty values that are otherwise arbitraryp. 123
Diagonal Lemmasp. 124
Selection Theoremsp. 127
A Selection Theorem for Lower Semi-continuous Set-valued Mapsp. 138
Examplep. 140
Remarksp. 144
Further applicationsp. 147
p. 149
Duals of Asplund Spacesp. 151
A Partial Converse to Theorem 5.4p. 156
Remarksp. 159
Bibliographyp. 161
Indexp. 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: 1st July 2002
Country of Publication: US
Dimensions (cm): 24.49 x 16.36  x 1.96
Weight (kg): 0.48