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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... Calculus of Measurable Maps 310 8.3 Proof of the Characterization Theorem 319 8.4 Limits of Measurable Maps and Selections 322 8.5 Tangent Cones in Lebesgue Spaces 324 8.6 Integral of Set- Valued Maps 326 8.7 Proofs of the Convexity ...
Proof — The first statement is obvious. The second one is a consequence of the following more general result: Proposition 1.1.5 Let us consider sequences of subsets Ln and Mn of a metric space and assume that there exists a compact ...
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 ...
Kn) = A(a - Limsup^^Kn) Proof — Observe that the first statement follows from the second one and Proposition 1.2.2 i). Hence we have to prove only the second claim. Let us consider a sequence xn € Kn such that a subsequence of elements ...
Proof — Set y = f(x) and consider a sequence of elements yn £ Mn converging to y. ... x being the limit of xn € f~1(Mn) belongs to the lower limit of the inverse images of the subsets Mn. The proof of the second statement is analogous.