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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A generalized sequence is a map u e M i-> € X An element x € X is the limit of if, for every neighborhood V of x, there exists uq € M. such that xM belongs to V for all a >z uq. An element x is said to be a cluster point of this ...
Let y belong to K. Since y does not belong to Limsup^^Ln, there exist ey > 0 and Ny such that, for all n > Ny, y does not belong to B(Ln,ey). The subset K being compact, it can be covered by p balls B(yi,eVi). This implies that for all ...
... y) \yeMn *€Ln / y€M» *€Lb Proof — Let y belong to M". Since tp is upper semicontinuous, we know that for any £ > 0 and any zeL*, there exist neighborhoods N(z) of z and N(y) of y such that Vz'€JV(z), Vj/'e N(y), ip(z',y') < <p(z,y) ...
Then, for all n > N and y € M„, j/ belongs to some N(yi), so that, inf ip(z, y) < inf tp(z, yi) + e < sup inf <p(z, y) + s Hence we have proved that for any e > 0, there exists N > 0 such that V n > N, sup inf tp(z, y) < sup inf ip(z, ...
K'j'^ C X \ Um which contradicts the fact that xq belongs to Um. □ 1.1.3 The Duality Theorem For closed convex cones Kn, upper limits and lower limits can be exchanged by duality. We introduce the (negative) polar cones to subsets K C ...