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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On the other hand, W:=WUV being a neighborhood of M, we deduce from the assumption that there exists Ni such that Vn > JVi, Mn C WU V Therefore LnC\Mn <zU for all n > max(iVo, N\). □ Remark — If M is not compact, but just closed, ...
Let us fix an open subset Um such that xq e Um C U. We thus deduce that Um n (Limsup^00.Dn.) = 0 Since for rij > m, Dnj :— = KpTM~1^ for some pj, we observe that Dn. is a subsequence of the sequence f K^1 ^) , the upper 3 V / n>0 limit ...
... (x + eB) D Kn = 0 The Separation Theorem implies the existence of elements pn G X* of norm equal to one such that °~Kn(Pn) < <Pn,X> -£\\Pn\\ = < Pn, X > -£ The subsets Kn being cones, we deduce that pn G K~ and that °~Kn(Pn) = 0.
... 2 — The square product HOG : X ~> Z of H and G at x is defined by (HaG)(x) := pi H(y) Let 1 denote the identity map from one set to itself. We deduce the following formulas f Graph(HoG) = (G x l)'1 (Gmph(H)) = (1 x tf)(Graph(G9) ...
Indeed, let V C Y be any open subset such that F(x) D V ^ 0. Then there exists n € N and y G F(x)nVW C F(x)C\V Thus x E Ln and therefore it belongs to Int(Z/„.) Since Ln = F-^K) C F-^V) we deduce that F is lower semicontinuous at x.