Title: On The Integral Representation of Strictly Continuous Set-Valued Maps
Author(s): Anaté K. Lakmon, Kenny K. Siggini
Pages: 114-120
Cite as:
Anaté K. Lakmon, Kenny K. Siggini, On The Integral Representation of Strictly Continuous Set-Valued Maps, Int. J. Anal. Appl., 9 (2) (2015), 114-120.


Let T be a completely regular topological space and C(T) be the space of bounded, continuous real-valued functions on T. C(T) is endowed with the strict topology (the topology generated by seminorms determined by continuous functions vanishing at in_nity). R. Giles ([13], p. 472, Theorem 4.6) proved in 1971 that the dual of C(T) can be identi_ed with the space of regular Borel measures on T. We prove this result for positive, additive set-valued maps with values in the space of convex weakly compact non-empty subsets of a Banach space and we deduce from this result the theorem of R. Giles ([13], theorem 4.6, p.473).

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  1. R. C. Buck, Bounded continuous functions on locally compact space, Michigan Math. J. 5 (1958), 95–104. Google Scholar

  2. R. C. Buck, operator algebras and dual spaces, Proc. Amer. Math. Soc. 3.681– 687 (1952). Google Scholar

  3. BOURBAKI, El´ements de Maths. Livre VI Int´egration-Chp. IX, Ed. Hermann, Paris 1969. Google Scholar

  4. A. Choo, strict topology on spaces of continuous vector-valued functions, Canad. J. Math. 31 (1979), 890–896. Google Scholar

  5. A. Choo, Separability in the strict topology, J. Math. Anal. Appl. 75 (1980), 219–222. Google Scholar

  6. H. S Collins, On the space l∞(S), with the stict topology, Math. Zeitschr. 106, 361–373 (1968). Google Scholar

  7. H. S. Collins and J. R. Dorroh, Remarks on certain function spaces, Math. Ann., 176, (1968), 157–168 . Google Scholar

  8. J. B. Conway, the strict topology and compactness in the space of measures, Bull. Amer. Math. Soc. 72, (1966), 75–78 . Google Scholar

  9. , J. Diestel, Sequences and Series in Banach spaces, Graduate Texts in Math., vol.92, Springer-Verlag, 1984. Google Scholar

  10. Drewnowsky, Topological Rings of Sets, Continuous Set Functions, Integration. III, Bull. Acad. Polon. Sci., S´er. Sci. Math., Astronom. et Phys., 20 (1972), 441–445. Google Scholar

  11. N. Dunford and J. Schwartz, Linear operators Part I, New York: Interscience 1958. Google Scholar

  12. R. A. Fontenot, Strict topologies for vector-valued functions, Canadian. J. Math. 26 (1974), 841–853. Google Scholar

  13. R. Giles, A Generalization of the Strict Topology, Trans. Amer. Math. Soc. 161(1971), 467–474. Google Scholar

  14. D. Gulick, The σ-compact-open topology and its relatives, Math. Scand.. 30 (1972), 159–176. Google Scholar

  15. J. Hoffman-J¨orgenson, A generalization of strict topology, Math. Scand. 30 (1972), 313–323. Google Scholar

  16. L. H¨ormander, Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Arkiv F¨or MATEMATIK. 3 nr 12 (1954). Google Scholar

  17. A. K. Katsaras, On the strict topology in the non-locally convex setting II, Acta. Math. Hung. 41 (1-2) (1983), 77–88. Google Scholar

  18. A. K. Katsaras, Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, Trans. Amer. Math. Soc. 216 (1979), 367–387. Google Scholar

  19. L. A. Khan, The strict topology on a space of vector-valued functions, Proc. Edinburgh Math. Soc., 22 (1979), 35–41. Google Scholar

  20. G. K¨othe, Topological Vector spaces I Second printing, Springer-Verlag, NewYork, 1983. Google Scholar

  21. R. Pallu De La Barriere, Publications Math´ematiques de l’Universit´e Pierre et Marie Curie No33. Google Scholar

  22. F. D. Sentilles, Bounded continuous functions on a completely regular space, Trans. Amer. Math. Soc., 168 (1972), 311–336. Google Scholar

  23. K. K. Siggini, Narrow Convergence in Spaces of Set-valued Measures, Bull. of The Polish Acad. of Sc. Math. Vol. 56, N◦1, (2008). Google Scholar

  24. K. K. Siggini, Sur les propri´et´es de r´egularit´e des mesures vectorielles et multivoques sur les espaces topologiques g´en´eraux, Th`ese de Doctorat de l’Universit´e de Paris 6. Google Scholar

  25. C. Todd, Stone-Weierstrass theorems for the strict topolgy, Proc. Amer. Math. Soc. 16 (1965), 657–659. Google Scholar

  26. A. C. M. Van Rooij, Tight functionals and the strict topology, Kyungpook Math.J.7 (1967), 41–43. Google Scholar

  27. J. Wells, Bounded continuous vector-valued functions on a locally compact space, Michigan Math. J. 12 (1965), 119–126. Google Scholar

  28. X. Xiaoping, C. Lixin, L. Goucheng, Y. Xiaobo, Set valued measures and integral representation, Comment.Math.Univ.Carolin. 37,2 (1996)269–284 Google Scholar


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