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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The first one states that a closed convex process defined on the whole space is continuous, and the second states that pointwise bounded families of closed convex processes are bounded — a prerequisite for studying the convergence of ...
The main difference with sequences of a metric space is that a converging generalized sequence is not necessarily bounded. (We can also replace generalized sequences by "filters.") The upper limit of a generalized sequence of subsets ...
The key facts that can be recalled now are that X* is still the dual of X supplied with the weakened topology, that X is the dual of X* supplied with the weak-* topology, and that the bounded subsets of the dual X* are weakly relatively ...
Lemma 1.1.9 Let us consider a sequence of subsets Kn contained in a bounded subset of a finite dimensional vector space X. Then co (Limsup^oo-Kn) = f] co I (J Kn N>0 \n>N Proof — The closed convex hull of the upper limit is obviously ...
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, ...