Uniform convergence of multiplier convergent series


  • Charles Swartz New Mexico State University.




If ? is a sequence K-space and Pxj is a series in a topological vector space X, the series is said to be ?-multiplier convergent if the series P? j=1 tjxj converges in X for every t = {tj} ? ?. We show that if ? satisfies a gliding hump condition, called the signed strong gliding hump condition, then the series P? j=1 tjxj converge uniformly for t = {tj} belonging to bounded subsets of ?. A similar uniform convergence result is established for a multiplier convergent series version of the Hahn-Schur Theorem.

Author Biography

Charles Swartz, New Mexico State University.

Mathematics Department. 


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

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

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

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

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

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

[St1] Stuart, C., Weak Sequential Completeness in Sequence Spaces, Ph.D. Dissertation, New Mexico State University, (1993).

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

[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).

[Sw1] Swartz, C., The Schur lemma for bounded multiplier convergent series, Math. Ann., 263, pp. 283-288, (1983).

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

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



How to Cite

C. Swartz, “Uniform convergence of multiplier convergent series”, Proyecciones (Antofagasta, On line), vol. 26, no. 1, pp. 27-35, Apr. 2017.




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