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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.28 n.1 Antofagasta mayo 2009

http://dx.doi.org/10.4067/S0716-09172009000100008 

Proyecciones Journal of Mathematics
Vol. 28, No 1, pp.89—109, May 2009.
Universidad Católica del Norte
Antofagasta - Chile


AN ABSTRACT GLIDING HUMP PROPERTY


CHARLES SWARTZ

New Mexico State University, U. S. A.

Correspondencia a:


Abstract
In this paper we introduce an abstract gliding hump property for sequence spaces which includes the signed weak and strong gliding hump properties as special cases. Further examples of sequence spaces satisfying the abstract gliding hump property are given, We then derive results concerning uniform convergence in β-duals, Hahn-Schur theorems and Orlicz-Pettis theorems for multiplier convergent series whose multiplier space satisfies the abstract gliding hump property.



REFERENCES
[Ap] T. Apostol, Mathematical Analysis, Addison-Wesley, Reading,         [ Links ] 1975.
[BP] C. Bessaga and A. Pelczynski, On Bases and Unconditional Convergence of Series in Banach Spaces, Studia Math., 17 (1958),15         [ Links ]1-164.
[Bo] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford ,         [ Links ]2000.
[BSS] J. Boos, C. Stuart and C. Swartz, Gliding hump properties of matrix domains, Analysis Math., 30 (2004), 24         [ Links ]3-257.
[DU] J. Diestel and J. Uhl, Vector Measures, Amer. Math. Soc., Providence,         [ Links ] 1977.
[Di] N. Dinculeanu, Weak Compactness and Uniform Convergence of Operators in Spaces of Bochner Integrable Functions, J. Math. Anal. Appl., 109 (1985), 37         [ Links ]2-387.
[Ds] N. Dunford and J. Schwartz, Linear Operators I, Interscience, N.Y.,         [ Links ]1958.
[E] R. E. Edwards, Functional Analysis, Holt-Rinehart-Winston, N.Y.,         [ Links ] 1965.
[FL] W. Filter and I. Labuda, Essays on the Orlicz-Pettis Theorem I, Real. Anal. Exchange, 16 (1990/91), 39         [ Links ]3-403.
[Ka] N. Kalton, The Orlicz-Pettis Theorem, Contemporary Math., Amer. Math. Soc., Providence,         [ Links ] 1980.
[KG] K.G. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N.Y.,         [ Links ]1981.
[Kö] G. Köthe, Topological Vector Spaces, Springer-Verlag, Berlin,         [ Links ] 1969.
[LW] E. Lacey and R. Whitley, Conditions under which all the Bounded Linear Maps are Compact, Math. Ann., 158 (1965), 1-5,         [ Links ](1965).
[LS] Li, Ronglu and C. Swartz, An Abstract Orlicz-Pettis Theorem and Applications, Proy. J. Math., 27, pp. 155-169,         [ Links ](2008).
[LWZ] Li, Ronglu and Wang, Fubin and Zhong, S., The strongest intrinsic meaning of sequential-evaluation convergence, Topology and its Appl., 154, pp. 1195-1205,         [ Links ](2007).
[MR] C. W. McArthur and J. Retherford, Some Applications of an Inequality in Locally Convex Spaces, Trans. Amer. Math. Soc., 137, pp. 115-123;         [ Links ](1969).
[MM] G. Metafune and V. B. Moscatelli, On the Space , Math. Nachr., 147, pp. 7-12,         [ Links ](1990).
[No] D. Noll, Sequential Completeness and Spaces with the Gliding Hump Property, Manuscripta Math., 66, pp. 237-252,         [ Links ](1990).
[Or] W. Orlicz, Beitrage zur Theorie der Orthogalent wichlungen II, Studia Math., 1, pp. 241-255,         [ Links ](1929).
[Pe] B. J. Pettis, Integration in Vector Spaces, Trans. Amer. Math. Soc. 44, pp. 277-304,         [ Links ](1938).
[St] C. Stuart, Weak Sequential Completeness of ß-duals, Rocky Mountain Math. J., 26, pp. 1559-1568,         [ Links ](1996).
[StSw1] C. Stuart and C. Swartz, Uniform Convergence in the Dual of a Vector-Valued Sequence Space, Taiwanese J. Math., 7, pp. 665-676,         [ Links ](2003).
[StSw2] C. Stuart and C.Swartz, Generalizations of the Orlicz-Pettis Theorem, Proy. J. Math., 24, pp. 37-48,         [ Links ](2005).
[Sw1] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N. Y.,         [ Links ](1992).
[Sw2] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ. Singapore,         [ Links ](1996).
[Sw3] C. Swartz, The Schur and Hahn Theorems for Operator Matrices, Rocky Mountain Math. J., 15, pp. 61-73,         [ Links ](1985).
[Sw4] C. Swartz, A Multiplier Gliding Hump Property for Sequence Spaces, Proy. J. Math., 20, pp. 19-31,         [ Links ](2001).
[Sw5] C. Swartz, Orlicz-Pettis Theorems for Multiplier Convergent Operator Valued Series, Proy. J. Math., 23, pp. 61-72,         [ Links ](2004).
[Sw6] C. Swartz, Uniform Convergence of Multiplier Convergent Series, Proy J. Math., 26, pp. 27-35,         [ Links ](2007).
[Sw7] C. Swartz, Interchanging Orders of Summation for Multiplier Convergent Series, Bol. Soc. Mat. Mexicana (3) 8, pp. 31-35,         [ Links ](2002).
[Wi] Wilansky, Modern Methods in Topological Vector Spaces, McGraw- Hill, N. Y.,         [ Links ](1978).
[ZLY] hong, S. and Li, Ronglu and Yang, Hong, Summability Results for Matrices of Quasi-Homogeneous Operators, Proy. J. Math., 27, pp. 249-258,         [ Links ](2008).

CHARLES SWARTZ
Department of Mathematics
New Mexico State University
Las Cruces, NM 88003
U. S. A.
e-mail : cswartz@nmsu.edu

Received : December 2008. Accepted : March 2009