Abstract

We show that if a Banach space XX is weakly compactly generated and CC, CnC_n are weak-star-closed bounded convex nonempty subsets of the dual space X∗X^*, then the support functionals δCn∗\delta^*_{C_n} converge to δC∗\delta^*_C pointwise on XX if and only if the sequence (Cn)(C_n) is uniformly bounded with weak-star limit CC.

Author information

Contact details are reproduced from the original publication and may be historical.

Simon P. Fitzpatrick

Keywords

  • Scalar convergence
  • weak-star convergence
  • set convergence
  • weakly compactly generated

References

  1. D. Amir, J. Lindenstrauss: The structure of weakly compact sets in Banach spaces, Ann. Math. 88 (1968) 35–46.https://doi.org/10.2307/1970554Google Scholar
  2. A. Balayadi, C. Zalinescu: Bounded scalar convergence, J. Math. Anal. Appl. 193 (1995) 134–157.https://doi.org/10.1006/jmaa.1995.1226Google Scholar
  3. G. Beer: Support and distance functionals for convex sets, Numer. Funct. Anal. Optimization 10 (1989) 15–36.https://doi.org/10.1080/01630568908816288Google Scholar
  4. F. S. De Blasi, J. Myjak: Weak convergence of convex sets in Banach spaces, Arch. Math. 47 (1986) 448–456.https://doi.org/10.1007/BF01189987Google Scholar
  5. J. M. Borwein, S. Fitzpatrick: Mosco convergence and the Kadec property, Proc. Am. Math. Soc. 106 (1989) 843–849.https://doi.org/10.1090/S0002-9939-1989-0969313-4Google Scholar
  6. W. J. Davis, T. Figiel, W. B. Johnson, A. Pelczynski: Factoring weakly compact operators, J. Funct. Anal. 17 (1974) 311–327.https://doi.org/10.1016/0022-1236(74)90044-5Google Scholar
  7. J. Diestel: Geometry of Banach Spaces. Selected Topics, Lecture Notes in Mathematics 485, Springer, Berlin (1975).https://doi.org/10.1007/BFb0082079Google Scholar
  8. J. Diestel: Sequences and Series in Banach Spaces, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5200-9Google Scholar
  9. J. Diestel, J. J. Uhl: Vector Measures, American Mathematical Society, Providence (1977).https://doi.org/10.1090/surv/015Google Scholar
  10. R. B. Holmes: Geometric Functional Analysis and its Applications, Springer, New York (1975).https://doi.org/10.1007/978-1-4684-9369-6Google Scholar
  11. J.-L. Joly: Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pur. Appl., IX. Sér. 52 (1973) 421–441.Google Scholar
  12. U. Mosco: Convergence of convex sets and solutions of variational inequalities, Adv. Math. 3 (1969) 510–585.https://doi.org/10.1016/0001-8708(69)90009-7Google Scholar
  13. U. Mosco: On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971) 518–539.https://doi.org/10.1016/0022-247X(71)90200-9Google Scholar
  14. R. Robert: Convergence de fonctionelles convexes, J. Math. Anal. Appl. 45 (1974) 533–555.https://doi.org/10.1016/0022-247X(74)90050-XGoogle Scholar
  15. W. Rudin: Functional Analysis, McGraw-Hill, New York (1973).Google Scholar
  16. G. Salinetti, R. J.-B. Wets: On the convergence of convex sets in finite dimensions, SIAM Rev. 21 (1979) 18–33.https://doi.org/10.1137/1021002Google Scholar
  17. Y. Sonntag: Convergence des Suites d’Ensembles, Monograph, Université de Provence, Marseille (1987).Google Scholar
  18. Y. Sonntag, C. Zalinescu: Scalar convergence of convex sets, J. Math. Anal. Appl. 164 (1992) 219–241.https://doi.org/10.1016/0022-247X(92)90154-6Google Scholar
  19. Y. Sonntag, C. Zalinescu: Set convergences. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993) 199–226.https://doi.org/10.1090/S0002-9947-1993-1173857-8Google Scholar
  20. K. R. Stromberg: An Introduction to Classical Real Analysis, Wadsworth, Belmont (1981).Google Scholar
  21. R. A. Wijsman: Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70 (1964) 186–188.https://doi.org/10.1090/S0002-9904-1964-11072-7Google Scholar
  22. R. A. Wijsman: Convergence of sequences of convex sets, cones and functions II, Trans. Amer. Math. Soc. 123 (1966) 32–45.https://doi.org/10.1090/S0002-9947-1966-0196599-8Google Scholar

Suggested citation

S. P. Fitzpatrick, A. S. Lewis. “Weak-Star Convergence of Convex Sets.” Journal of Convex Analysis 13 (2006), No. 3/4, 711–719. https://doi.org/10.68381/jca13053

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.