Set-Valued Analysis"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." —Mathematical Reviews "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." —Zentralblatt Math |
From inside the book
Results 1-5 of 80
... 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 generalized sequence if, for every neighborhood V of a; and every u € M, there exists v > u such that xu e V. We recall that a ...
... exists an integer N such that Ln n Mn C U whenever n> N. Proof — If the neighborhood U contains M, the result ... 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 ...
... 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, the conclusion of the proposition remains true for any neighborhood U of M n (Limsup^ooLjj) whose ...
... exists an integer No such that Vn>iV0, Mn C [J N(yi) i=l,...,n Set JV := maxj=o,...,n iVyi. 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 ...
... exist xo € Limsup^oo-Dn & xo £ Limmfn-^ooAi The latter condition means that there exists an open neighborhood U of xq and a subsequence Dnj such that U D Dnj = 0 for any j. Let us fix an open subset Um such that xq e Um C U. We thus ...