Uniform Convergence and the Hahn-Schur Theorem

Charles Swartz

New Mexico State University, U.S.A.

ABSTRACT

Let E be a vector space, F aset, G be a locally convex space, b : E X F — G a map such that ò(-,y): E — G is linear for every y G F; we write b(x, y) = x · y for brevity. Let ë be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(-,y): E — G are continuous for all y G F .A series Xj in X is ë multiplier convergent with respect to w(E, F) if for each t = {tj} G ë ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces ë satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose _j XX ij is ë multiplier convergent with respect to w(E, F) for each i G N and for each t G ë the set {Xj=! tj Xj : i} is uniformly bounded on any subset B C F such that {x · y : y G B} is bounded for x G E.Then for each t G ë the series ^jjLi tj xj · y converge uniformly for y G B,i G N. This result is used to prove a Hahn-Schur Theorem for series such that lim¿ Xj=! tj xj · y exists for t G ë,y G F. Applications of these abstract results are given to spaces of linear operators, vector spaces in duality, spaces of continuous functions and spaces with Schauder bases.

Key Words : Multiplier convergent series, uniform convergence, Hahn-Schur Theorem.

REFERENCES

[CL] A. Chen, R. Li, A version of Orlicz-Pettis Theorem for quasi-homogeneous operator space, J. Math. Anal. Appl., 373, pp. 127-133,(2011).

[Ga] D. J. H. Garling, The â — and ã—duality of sequence spaces, Proc.Camb. Phil. Soc., 63, pp. 963-981, (1967).

[Ha] H. Hahn, Uber Folgen linearen Operationen, Monatsch. fur Math. und Phys., 32, pp. 1-88, (1922).

[K1] G. Kothe, Topological Vector Spaces I, Springer, Berlin, (1983). [K2] G. Kothe, Topological Vector Spaces II, Springer, Berlin, (1979). [LW] R. Li, J. Wang, Invariants in Abstract Mapping Pairs, J. Aust. Math.Soc., 76, pp. 369-381, (2004).

[Sc] J. Schur, Uber lineare Transformation in der Theorie die unendlichen Reihen, J. Reine Agnew. Math., 151, pp. 79-111, (1920).

[St] W. J. Stiles, On Subseries Convergence in F-spaces, Israel J. Math., 8, pp. 53-56, (1970).

[Sw1] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N. Y., (1992).

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

[Sw3] C. Swartz, Multiplier Convergent Series, World Sci. Publ., Singapore, (2009).

[Th] G. E. F. Thomas, L'integration par rapport a une mesure de Radon vectorielle, Ann. Inst. Fourier, 20(, pp. 55-191, (1970).

[Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, N. Y., (1978).

Received : January 2012. Accepted : February 2012

Charles Swartz

Department of Mathematical Sciences

New Mexico State University

Las Cruces, NM 88003, U. S. A.

e-mail : cswartz@nmsu.edu