## 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 32

when N := {a>i, . . . , xn} ranges over the finite subsets of X. In other words, a generalized sequence of elements xp ... 5This means that there is countable basis of

**open subsets**, or, equivalently, that there exists a countable basis ...

... there exists an integer N such that Ln n Mn C U whenever n> N. Proof — If the neighborhood U contains M, the result follows from the assumption on M. Otherwise, by taking an

**open**neighborhood U, the

**subset**K := M\U is not empty, ...

Proof — Since X is separable, there exists a countable family of

**open subsets**Um satisfying the following property: V

**open subset**U, V x € U, 3 Um such that x € Um C U Let us consider a sequence of subsets Kn. We shall construct a ...

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 ...

We can prove a similar result for non convex subsets and nonlinear maps, which is based on extensions of Graves' Inverse Function Theorem 3.4.2 proved in Chapter 3. It states that if f is a continuous map from an

**open subset**of a Banach ...

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