Springer Science & Business Media, Mar 2, 2009 - Science - 461 pages
"An elegantly written, introductory overview of the field, with a near perfect choice of what to include and what not, enlivened in places by historical tidbits and made eminently readable throughout by crisp language. It has succeeded in doing the near-impossible—it has made a subject which is generally inhospitable to nonspecialists because of its ‘family jargon’ appear nonintimidating even to a beginning graduate student."
—The Journal of the Indian Institute of Science
"The book under review gives a comprehensive treatment of basically everything in mathematics that can be named multivalued/set-valued analysis. It includes...results with many historical comments giving the reader a sound perspective to look at the subject...The book is highly recommended for mathematicians and graduate students who will find here a very comprehensive treatment of set-valued analysis."
"I recommend this book as one to dig into with considerable pleasure when one already knows the subject...‘Set-Valued Analysis’ goes a long way toward providing a much needed basic resource on the subject."
—Bulletin of the American Mathematical Society
"This book provides a thorough introduction to multivalued or set-valued analysis...Examples in many branches of mathematics, given in the introduction, prevail [upon] the reader the indispensability [of dealing] with sequences of sets and set-valued maps...The style is lively and vigorous, the relevant historical comments and suggestive overviews increase the interest for this work...Graduate students and mathematicians of every persuasion will welcome this unparalleled guide to set-valued analysis."
Results 1-5 of 83
9 Selections and Parametrization 353 9.1 Case of lower semicontinuous maps 355 9.2 Case of upper semicontinuous maps 358 9.3 Minimal Selection 360 9.4 The Steiner Selection 364 9.4.1 Steiner Points of Convex Compact Sets .
It happens that set-valued maps with closed graph taking their values in a compact set are upper semicontinuous. Hence we have an easy way to verify upper semicontinuity of a map. Lower semicontinuity is not as simple to check: the last ...
... we observe that Bx(K,r) = K + rBx The balls B(K,r) are neighborhoods of K. When K is compact, each neighborhood of K contains such a ball around K. Limits of sets have been introduced by Painleve3 in 1902, as it is reported by his ...
... sequence converging to x to a generalized sequence converging to f(x) and that a subset K C X is compact if and only if every generalized sequence of elements of K has a cluster point. (See for instance [148, Section 1.7].) ...
We could for instance use the fact that the weakly compact subsets of the dual X* of a separable5 Banach space are metrizable (for the weak-* topology) (see [148, Theorem 5.6.3]) to avoid generalized sequences.