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

Abstract

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

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

Dates
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1453817252

Digital Object Identifier
doi:10.1216/RMJ-2015-45-5-1511

Mathematical Reviews number (MathSciNet)
MR3452226

Zentralblatt MATH identifier
1352.46018

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

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

Citation

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. https://projecteuclid.org/euclid.rmjm/1453817252


Export citation

References

  • R.D. Bourgin, Geometric aspects of convex sets with the Radon-Nikod$\acute{y}$m property, Lect. Notes in Math., 993, Springer, Berlin, 1983.
    Mathematical Reviews (MathSciNet): MR704815
  • A. Brøndsted and R.T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611.
    Mathematical Reviews (MathSciNet): MR178103
    Zentralblatt MATH: 0141.11801
    Digital Object Identifier: doi:10.1090/S0002-9939-1965-0178103-8
  • J. Diestel, and J.J. Uhl, Jr., Vector measures, American Mathematical Society Surveys 15, Providence, Rhode Island, 1977.
    Mathematical Reviews (MathSciNet): MR453964
  • L. Drewnowski and I. Labuda, On minimal convex usco and maximal monotone maps, Real Anal. Exch. 15 (1989/90), 729–742.
    Mathematical Reviews (MathSciNet): MR1059434
    Zentralblatt MATH: 0716.54015
  • V. Farmaki, A geometric characterization of the weak Radon-Nikodým property in dual Banach spaces, Rocky Mountain J. Math. 25 (1995), 611–617.
    Mathematical Reviews (MathSciNet): MR1336553
    Digital Object Identifier: doi:10.1216/rmjm/1181072240
    Project Euclid: euclid.rmjm/1181072240
  • R.C. James, A separable somewhat reflexive Banach spaces with nonseparable dual, Bulletin Amer. Math. Soc. 80 (1974), 738–743.
    Mathematical Reviews (MathSciNet): MR417763
    Zentralblatt MATH: 0286.46018
    Digital Object Identifier: doi:10.1090/S0002-9904-1974-13580-9
    Project Euclid: euclid.bams/1183535714
  • M. Lassonde, Aspund spaces, Stegall variational principle and the RNP, Set-Valued Anal. 17 (2009), 183–193.
    Mathematical Reviews (MathSciNet): MR2529695
    Zentralblatt MATH: 1176.46024
    Digital Object Identifier: doi:10.1007/s11228-009-0111-6
  • 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.
    Mathematical Reviews (MathSciNet): MR390720
  • K. Musial, The weak Radon-Nikodým property in Banach space, Stud. Math. 64 (1978), 151–174.
    Mathematical Reviews (MathSciNet): MR537118
  • A. Pelczynski, On Banach spaces containing $L_{1}$, Stud. Math. 30 (1968), 231–246.
    Mathematical Reviews (MathSciNet): MR232195
  • R.R. Phelps Convex functions, monotone operators and differentiability, Lect. Notes Math. 1364 (1993), Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1238715
  • H.P. Rosenthal, Characterization of Banach spaces containing $\ell_{1}$, Proc. Nar. Acad. Sci. 71 (1974), 2411–2413.
    Mathematical Reviews (MathSciNet): MR358307
    Zentralblatt MATH: 0297.46013
    Digital Object Identifier: doi:10.1073/pnas.71.6.2411
  • E. Saab and P. Saab, Some characterizations of weak Radon-Nikodým sets, Proc. Amer. Math. Soc. 86 (1982), 307–311.
    Mathematical Reviews (MathSciNet): MR667295
  • ––––, A dual geometric characterization of Banach spaces not containing $\ell_{1}$, Pac. J. Math. 105 (1983), 415–425.
    Mathematical Reviews (MathSciNet): MR691612
    Zentralblatt MATH: 0465.46009
    Digital Object Identifier: doi:10.2140/pjm.1983.105.415
    Project Euclid: euclid.pjm/1102723337
  • ––––, Sets with the weak Radon-Nikodým property in dual Banach spaces, Ind. Univ. Math. J. 32 (1983), 527–541.
    Mathematical Reviews (MathSciNet): MR703283
    Digital Object Identifier: doi:10.1512/iumj.1983.32.32038

Rocky Mountain Mathematics Consortium

Rocky Mountain Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more