Proyecciones Journal of Mathematics
Vol. 26, No1, pp. 27-35, May 2007.
Universidad Católica del Norte
Antofagasta - Chile


UNIFORM CONVERGENCE OF MULTIPLIER CONVERGENT SERIES

CHARLES SWARTZ

NEW MEXICO STATE UNIVERSITY

Correspondencia a:

Abstract

If λ is a sequence K-space and Σ x_j is a series in a topological vector space X_; the series is said to be λ-multiplier convergent if the series converges in X for every t = {tj} λ. We show that if λ satisfies a gliding hump condition, called the signed strong gliding hump condition, then the series converge uniformly for t = {tj} belonging to bounded subsets of λ. A similar uniform convergence result is established for a multiplier convergent series version of the Hahn-Schur Theorem.

REFERENCES

[AP] Aizpuru, A. and Perez-Fernandez, J., Spaces of S-bounded multiplier convergent series, Acta. Math. Hungar., 87, pp. 135-146, (2000).         [ Links ]

[B] Boos, J., Classical and modern methods in summability, Oxford University Press, Oxford, (2000).         [ Links ]

[BSS] Boos, J.; Stuart, C.; Swartz, C, Gliding hump properties of matrix domains, Analysis Math., 30, pp. 243-257, (2004).         [ Links ]

[D] Day, M., Normed Linear Spaces, Springer-Verlag, Berlin, (1962).         [ Links ]

[FP] Florencio, M. and Paul, P., A note on A-multiplier convergent series, Casopis Pro Pest. Mat., 113, pp. 421-428, (1988).         [ Links ]

[N] Noll, D., Sequential completeness and spaces with the gliding hump property,Manuscripta Math., 66, pp. 237-252, (1960).         [ Links ]

[Stl] Stuart, C., Weak Sequential Completeness in Sequence Spaces, Ph.D. Dissertation, New Mexico State University, (1993).         [ Links ]

[St2] Stuart, C, Weak Sequential Completeness of /3-duals, Rocky Mountain Math. J., 26, pp. 1559-1568, (1996).         [ Links ]

[SS] Stuart, C. and Swartz, C, Uniform convergence in the dual of a vector-valued sequence space,Taiwanese J. Math., 7, pp. 665-676, (2003).         [ Links ]

[Swl] Swartz, C, The Schur lemma for bounded multiplier convergent series, Math. Ann., 263, pp. 283-288, (1983).         [ Links ]

[Sw2] Swartz, C, Infinite Matrices and the Gliding Hump, World Sci. Press, Singapore, (1996).         [ Links ]

[WCC] [WCC] Wu, J. ; Li, L. ; Cui, C, Spaces of A-multiplier convergent series, Rocky Mountain Math. J., 35, pp. 1043-1057, (2005).         [ Links ]

Charles Swartz
Mathematics Department
New Mexico State University
Las Cruces NM 88003USA
e-mail : cswartz@nmsu.edu