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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1.1 Limits of Sets 1.1.1 Definitions Let X be a metric space supplied with a distance d. When K is a subset of X, we denote by dx(x) := d(x,K) :— inf d(x,y) y&K 2These concepts should not be confused with the upper and lower ...
They have been popularized by Kuratowski in his famous book Topologie and thus, often called Kuratowski lower and upper limits of sequences of sets. Definition 1.1.1 Let (-Kn)neN be a sequence of subsets of a metric space X. We say that ...
It is easy to check that: Proposition 1.1.2 If (-Kn)neN *s a sequence of subsets of a metric space, then Liminf„_,.oo^n is the set of limits of sequences xn G Kn and Limsupn^oo-in is the set of cluster points of sequences ...
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 space by neighborhoods and the sequences of a ...
However, in general X is a closed subspace of the bidual X** of X, which is the dual of the Banach space X*, endowed with the norm ... we can present lower and upper limits in the framework of metric spaces or of (countable) sequences, ...