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."
Results 1-5 of 5
The upper limit is also equal to the subset of cluster points of "approximate"
sequences satisfying: Ve>0, 3N(e) such that V n > N(e), xn € B(Kn,e) Remark —
Replacing the balls of a metric space by neighborhoods and the sequences of a
... N>0n>N e>0N>0n>N and Liminfn^ooKn = p| (J p| B(Kn,e) £>0N>0 n>N and
single out the following property of upper limits: Theorem 1.1.4 Let K be a subset
of a metric space X satisfying the following property: for any neighborhood U of K,
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, the
Remark — We can use the concepts of inverse images and cores to characterize
upper and lower semicontinuous maps: Proposition 1.4.4 A set-valued map F : X
~» Y is upper semicontinuous at x if the core of any neighborhood of F(x) is a ...
Since / is upper semicontinuous, we can associate with any y € F(x) open
neighborhoods V(y) of y and Uy(x) of x such ... F(x) is compact, it can be covered
by n neighborhoods V(yi), i = 1, • • • ,p, the union of which makes up a
neighborhood of ...