Strongly Bounded Partial Sums


  • Charles Swartz New Mexico State University.



Orlicz-Pettis Theorem, locally convex spaces, convergence, teorema de Orlicz-Pettis, espacios localmente convexos, convergencia.


If λ is a scalar sequence space, a series P Zj in a topological vector space Z is λ multiplier convergent in Z if the series P ∞J =1 tj Zj converges in Z for every t = {tj} ∈ λ-If λ satisfies appropriate conditions, a series in a locally convex space X which is λ multiplier convergent in the weak topology is λ multiplier convergent in the original topology ofthe space (the Orlicz-Pettis Theorem) but may fail to be λ multiplier convergent in the strong topology of the space. However, we show under apprpriate conditions on the multiplier space λ that the series will have strongly bounded partial sums.

Author Biography

Charles Swartz, New Mexico State University.

Department of Mathematics.


[1] G. Bennett, A New Class of Sequence Spaces with Applications to Summability Theory, J. Reine Angew. Math., 266, pp. 49-75, (1974).

[2] J. Boos, Classical and Modern Methods in Summability, Oxford University Presss, Oxford, (2000).

[3] J. Boos, C. Stuart, C. Swartz, Gliding Hump Properties and Matrix Domains, Analysis Math., 30, pp. 243-257, (2004).

[4] P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N. Y., (1981).

[5] G. Kothe, Topological Vector Spaces I, Springer Verlag, Berlin, (1969).

[6] C. W. McArthur, On a Theorem of Orlicz and Pettis, Pacific J. Math., 22, pp. 297-303, (1967).

[7] D. Noll, Sequential Completeness and Spaces with the Gliding Humps Property, Manuscripta Math., 66, pp. 237-252, (1990).

[8] C. Stuart, Weak Sequential Completeness of β-Duals, Rocky Mountain Math. J., 26, pp. 1559-1568, (1996).

[9] C. Swartz, Multiplier Convergent Series, World Sci. Publ., Singapore, (2009).

[10] C. Swartz, An Abstract Gliding Hump Property, Proy.J. Math., 28, pp. 89-109, (2009).

[11] A. Wilansky, Modern Methods in Topological Vector Spaces, McGrawHill, N. Y., (1978).



How to Cite

C. Swartz, “Strongly Bounded Partial Sums”, Proyecciones (Antofagasta, On line), vol. 33, no. 2, pp. 205-213, Mar. 2017.




Similar Articles

You may also start an advanced similarity search for this article.