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."
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Any decreasing sequence of subsets Kn has a limit, which is the intersection of their closures: if Kn C Km when n > m, then Limn->oo^n — f") Kn n>0 An upper limit may be empty (no subsequence of elements xn G Kn has a cluster point.) ...
i.e., of limits of subsequences xn> € Kni. The upper limit is also equal to the subset of cluster points of "approximate" sequences satisfying: Ve>0, 3N(e) such that V n > N(e), xn € B(Kn,e) Remark — Replacing the balls of a metric ...
We shall say that the subset a - Limsup^ooifn of weak-* limits of subsequences of elements xn € Kn is the sequentially weak upper limit of the subsets Kn. In this way, we can present lower and upper limits in the framework of metric ...
For m — 0, we set Kn^ '□= Kn. Assume that the m — 1 first subsequences , 0 < p < m — 1 have been constructed. Consider the mth open subset Um. Then either for every subsequence Uj, ^□(Limsup^X^-1)) ^ 0 in which case we set K^1 ...
in which case we set K^1 := *\ (The choice of such a subsequence does not matter.) These sequences (k^\ being constructed, we extract the V / n>0 diagonal subsequence Dn := Kn*\ We claim that it has a set limit.