Rocky Mountain Journal of Mathematics

New criteria for the weak Radon-Nikodým property related to set-valued operators

Keun Young Lee

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we deal with new equivalent conditions for the localized weak Radon-Nikod\'ym property in dual Banach space related to set-valued operators. First, we introduce the geometric definition of the weak Radon-Nikod\'ym property and the weakly fragmented set-valued operator. Next, using the weakly fragmented mapping, we reveal the relation between the weak Radon-Nikod\'ym property and the weakly single-valued operator. Finally, using this relation and the concept of the exposed point, the main theorem is given together with some applications.

Article information

Rocky Mountain J. Math., Volume 45, Number 5 (2015), 1511-1526.

First available in Project Euclid: 26 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]
Secondary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

Radon-Nikodým weakly fragmented set-valued operator weakly single-valued operator $x^{**}$-weak$^{*}$ exposed point


Lee, Keun Young. New criteria for the weak Radon-Nikodým property related to set-valued operators. Rocky Mountain J. Math. 45 (2015), no. 5, 1511--1526. doi:10.1216/RMJ-2015-45-5-1511.

Export citation


  • R.D. Bourgin, Geometric aspects of convex sets with the Radon-Nikod$\acute{y}$m property, Lect. Notes in Math., 993, Springer, Berlin, 1983.
  • A. Brøndsted and R.T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611.
  • J. Diestel, and J.J. Uhl, Jr., Vector measures, American Mathematical Society Surveys 15, Providence, Rhode Island, 1977.
  • L. Drewnowski and I. Labuda, On minimal convex usco and maximal monotone maps, Real Anal. Exch. 15 (1989/90), 729–742.
  • V. Farmaki, A geometric characterization of the weak Radon-Nikodým property in dual Banach spaces, Rocky Mountain J. Math. 25 (1995), 611–617.
  • R.C. James, A separable somewhat reflexive Banach spaces with nonseparable dual, Bulletin Amer. Math. Soc. 80 (1974), 738–743.
  • M. Lassonde, Aspund spaces, Stegall variational principle and the RNP, Set-Valued Anal. 17 (2009), 183–193.
  • J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain $\ell_{1}$ and whose duals are non-separable, Stud. Math. 54 (1975), 81–105.
  • K. Musial, The weak Radon-Nikodým property in Banach space, Stud. Math. 64 (1978), 151–174.
  • A. Pelczynski, On Banach spaces containing $L_{1}$, Stud. Math. 30 (1968), 231–246.
  • R.R. Phelps Convex functions, monotone operators and differentiability, Lect. Notes Math. 1364 (1993), Springer, Berlin.
  • H.P. Rosenthal, Characterization of Banach spaces containing $\ell_{1}$, Proc. Nar. Acad. Sci. 71 (1974), 2411–2413.
  • E. Saab and P. Saab, Some characterizations of weak Radon-Nikodým sets, Proc. Amer. Math. Soc. 86 (1982), 307–311.
  • ––––, A dual geometric characterization of Banach spaces not containing $\ell_{1}$, Pac. J. Math. 105 (1983), 415–425.
  • ––––, Sets with the weak Radon-Nikodým property in dual Banach spaces, Ind. Univ. Math. J. 32 (1983), 527–541.