Multifunctions: the Rudiments

  • Gerald Beer
Part of the Mathematics and Its Applications book series (MAIA, volume 268)


If f is a function from a set Y to a set X,then for each xX there may be several points of Y sent by f to x. Thus the inverse assignment x → {y ∈ Y: f(y) = x} is naturally multi-valued, and unless f is onto, some of the values of the inverse are empty. Only when f is a bijection will this assignment be a single-valued function in the usual sense. In this chapter we present the elements of the theory of functions that assign to points of a given set X subsets of a second set Y. Such functions are called set-valued functions or multifunctions. There are a number of set-valued functions that naturally arise in Banach space geometry, some of which we introduce here.


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Copyright information

© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Gerald Beer
    • 1
  1. 1.Department of Mathematics and Computer ScienceCalifornia State UniversityLos AngelesUSA

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