A multiplier gliding hump property for sequence spaces
DOI:
https://doi.org/10.4067/S0716-09172001000100002Abstract
We consider the Banach-Mackey property for pairs of vector spaces E and E0 which are in duality. Let A be an algebra of sets and assume that P is an additive map from A into the projection operators on E. We define a continuous gliding hump property for the map P and show that pairs with this gliding hump property and another measure theoretic property are Banach-Mackey pairs,i.e., weakly bounded subsets of E are strongly bounded. Examples of vector valued function spaces, such as the space of Pettis integrable functions, which satisfy these conditions are given.References
[B] S. Banach, Theorie des Operations Lineaires, Warsaw, (1932).
[BF] J. Boos and D. Fleming, Gliding hump properties and some applications,Int. J. Math. Math. Sci., 18, pp. 121-132, (1995).
[Dr] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems,Bull. Acad. Polon. Sci., 20, pp. 725- 731, (1972).
[FP] M. Florencio and P. Paul, Barrelledness conditions on certain vector valued sequence spaces, Arch. Math., 48, pp. 153-164, (1987).
[F] J. Fourie, Barrelledness Conditions on Generalized Sequence Spaces, South African J. Sci., 84, pp. 346-348, (1988).
[GKR] M.Gupta, P.K.Kamthan, and K.L.N. Rao, Duality in Certain Generalized Kothe Sequence Spaces, Bull. Inst. Math. Acad. Sinica, 5, pp. 285-298, (1977).
[Ha] H. Hahn, Uber Folgen linearen Operationen, Monatsch. fur Math. und Phys., 32, pp. 1-88, (1922).
[HT] E. Hellinger and O. Toeplitz, Grundlagen fur eine Theorie den unendlichen Matrizen. Math. Ann., 69, pp. 289-330, (1910).
[Hi] T. H. Hilldebrandt, On Uniform Boundedness of Sets of Functional Operations, Bull. Amer. Math Soc., 29, pp. 309-315, (1923).
[K] G. Kothe, Topological Vector Spaces, Springer-Verlag, Berlin, (1979).
[L] H. Lebesgue, Sur les integales singulieres, Ann. de Toulouse, 1, pp. 25-117, (1909).
[M] J. Mendoza, A barrelledness criteria for C0(E), Arch. Math., 40, pp. 156-158, (1983).
[N] D. Noll, Sequential Completeness and Spaces with the Gliding Hump Property, Manuscripta Math., 66, pp. 237-252, (1990).
[RR] K.P.S. Rao and M. Rao,Theory of Charges, Academic Press, N. Y., (1983).
[S] H.H. Schaefer, Topological Vector Spaces, MacMillan, N. Y., (1966).
[Sw1] C.
[Sw2] C. Swartz,Measure, Integration and Function Spaces, World Sci. Publ., Singapore, (1994).
[Sw3] C. Swartz, Infinite Matrices and the Gliding Hump,World Sci. Publ., Singapore, (1996).
[Sw4] C. Swartz, Topological Properties of the Space of Integrable Functions with respect to a Charge, Ricerche di Mat., to appear.
[Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).
[BF] J. Boos and D. Fleming, Gliding hump properties and some applications,Int. J. Math. Math. Sci., 18, pp. 121-132, (1995).
[Dr] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems,Bull. Acad. Polon. Sci., 20, pp. 725- 731, (1972).
[FP] M. Florencio and P. Paul, Barrelledness conditions on certain vector valued sequence spaces, Arch. Math., 48, pp. 153-164, (1987).
[F] J. Fourie, Barrelledness Conditions on Generalized Sequence Spaces, South African J. Sci., 84, pp. 346-348, (1988).
[GKR] M.Gupta, P.K.Kamthan, and K.L.N. Rao, Duality in Certain Generalized Kothe Sequence Spaces, Bull. Inst. Math. Acad. Sinica, 5, pp. 285-298, (1977).
[Ha] H. Hahn, Uber Folgen linearen Operationen, Monatsch. fur Math. und Phys., 32, pp. 1-88, (1922).
[HT] E. Hellinger and O. Toeplitz, Grundlagen fur eine Theorie den unendlichen Matrizen. Math. Ann., 69, pp. 289-330, (1910).
[Hi] T. H. Hilldebrandt, On Uniform Boundedness of Sets of Functional Operations, Bull. Amer. Math Soc., 29, pp. 309-315, (1923).
[K] G. Kothe, Topological Vector Spaces, Springer-Verlag, Berlin, (1979).
[L] H. Lebesgue, Sur les integales singulieres, Ann. de Toulouse, 1, pp. 25-117, (1909).
[M] J. Mendoza, A barrelledness criteria for C0(E), Arch. Math., 40, pp. 156-158, (1983).
[N] D. Noll, Sequential Completeness and Spaces with the Gliding Hump Property, Manuscripta Math., 66, pp. 237-252, (1990).
[RR] K.P.S. Rao and M. Rao,Theory of Charges, Academic Press, N. Y., (1983).
[S] H.H. Schaefer, Topological Vector Spaces, MacMillan, N. Y., (1966).
[Sw1] C.
[Sw2] C. Swartz,Measure, Integration and Function Spaces, World Sci. Publ., Singapore, (1994).
[Sw3] C. Swartz, Infinite Matrices and the Gliding Hump,World Sci. Publ., Singapore, (1996).
[Sw4] C. Swartz, Topological Properties of the Space of Integrable Functions with respect to a Charge, Ricerche di Mat., to appear.
[Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).
Published
2017-04-24
How to Cite
[1]
C. Swartz, “A multiplier gliding hump property for sequence spaces”, Proyecciones (Antofagasta, On line), vol. 20, no. 1, pp. 19-31, Apr. 2017.
Issue
Section
Artículos
-
Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.