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."
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.) ...
The upper limit is also equal to the subset of cluster points of "approximate" sequences satisfying: Ve>0, 3N(e) such that V ... An element x is said to be a cluster point of this generalized sequence if, for every neighborhood V of a; ...
We recall that a continuous single-valued map from a metric space X to a metric space Y is proper if and only if one of the following equivalent statements If f(xn) converges in Y, then xn has a cluster point or i) f maps closed subsets ...
Let us consider a sequence xn € Kn such that a subsequence of elements A(xn) (again denoted A(xn)) converges weakly to some y in Y. We shall check that (£n)neN has a weak cluster point, by showing that it is weakly bounded, and thus, ...
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