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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.4 Antofagasta dic. 2014 

Subseries convergence in abstract duality pairs

Min-Hyung Cho

Kum-Oh National Institute of Tech.


Li Ronglu

Harbin Institute of Tech.

P. R. C.

Charles Swartz

New Mexico State University



Let E, F be sets, G an Abelian topological group and b : ExF — G. Then (E, F, G) is called an abstract triple. Let w(F, E) be the weakest toplogy on F such that the maps {b(x, ·): x G E} from F into G are continuous. A subset B C F is w(F,E) sequentially conditionally compact if every sequence {yk} C B has a subsequence {ynk } such that limj; b(x, ynk) exists for every x G E. It is shown that if a formal series in E is subseries convergent in the sense that for every subsequence {xnj} there is an element x G E such that Xj=! b(xnj ,y) = b(x,y) for every y G F ,then the series Xj=! b(xnj ,y) converge uniformly for y belonging to w(F, E) sequentially conditionally compact subsets ofF. This result is used to establish Orlicz-Pettis Theorems in locall convex and function spaces. Applications are also given to Uniform Boundedness Principles and continuity results for bilinear mappings.


[AB] P. Antosik, Z. Bu, R. Li, E. Pap and C. Swartz, K -convergent sequences in locally convex spaces, Generalized Functions and Convergence, World Scientific, Singapore, pp. 299-304, (1990).         [ Links ]

[AS1] P. Antosik and C. Swartz, Matrix Methods in Analysis, Singer-Verlag, Heidelberg, (1985).         [ Links ]

[AS2] P. Antosik and C. Swartz, The Schur and Phillips Lemmas for Topological Groups, J. Math. Anal Appl., 98, pp. 179-187, (1984).

[B] B. Basit, On a Theorem of Gelfand and a new proof of the Orlicz-Pettis Theorem, Rend. Inst. Matem. Univ. di Trieste, 18 (1986), pp. 159-162, (1986).         [ Links ]

[BK] G. Bennett and N. Kalton, FK-spaces containing co, Duke Math. J., 39, pp. 561-582, (1972).

[C] C. Constantinescu, On Nikodyms Boundedness Theorem, Libertas Math., 1, pp. 51-73, (1981).         [ Links ]

[BCS] O. Blasco, J. M. Calabuig and T. Signes, A bilinear version of Orlicz-Pettis Theorem, J. Math. Anal. Appl., 348, pp. 150-164, (2008).         [ Links ]

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

[Diel] P. Dierolf, Theorems of Orlicz-Pettis type for locally convex spaces, Man. Math., 20, pp. 73-94, (1977).         [ Links ]

[DF] J. Diestel and B. Faires, On Vector Measures, Trans. Amer. Math. Soc., 198, pp. 253-271, (1974).

[DU] J. Diestel and J. Uhl, Vector Measures, Amer. Math. Soc. Surveys #15, Providence, (1977).         [ Links ]

[Di] N. Dinculeanu, Weak Compactness and Uniform Convergence of Operators in Spaces of Bochner Integrable Function, J. Math. Anal. Appl., 109 (1985).         [ Links ]

[Dr] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems, Bull. Aced. Polon. Sci., 20, pp. 725-731, (1972).         [ Links ]

[DS] N. Dunford and J. Schwartz, Linear Operators, Interscience, N.Y., (1958).         [ Links ]

[G] H. Garnir, M. DeWilde and J. Schmets, Analyse Fontionnelle I, Birkhauser, Basel, (1968).         [ Links ]

[FL] W. Filter and I. Labuda, Essays on the Orlicz Theeorem I, Real Anal. Exch., 16, pp. 393-403, (1990-1991).         [ Links ]

[K2] N. Kalton, Subseries Convergence in Topological Groups and Vector Spaces, Israel J. Math., 10, pp. 402-412, (1971).         [ Links ]

[K3] N. Kalton, Spaces of Compact Operators, Math. Ann., 208, pp. 267278, (1974).

[Kh] S. M. Khaleelulla, Counter Examples in Topological Vector Spaces, Springer Lecture Notes in Mathematics 936, Heidelberg, (1982).         [ Links ]

[LC] R.Li and M. Cho, A General Kalton-type Theorem, J. Harbin Inst. Tech 25, pp. 100-104, (1992).         [ Links ]

[LS1] R. Li and C. Swartz, Spaces for which the Uniform Boundedness Principle Holds, Studia Sci. Math. Hungar., 27, pp. 379-384, (1992).         [ Links ]

[LS2] R. Li and C. Swartz, A Nonlinear Schur Theorem, Acta Sci. Math. (Szged), 58, pp. 497-508, (1993).         [ Links ]

[LS3] R. Li and C. Swartz, An Abstract Orlicz-Pettis Theorem and Applications, Proy. J. Math., 27, pp. 155-169, (2008).         [ Links ]

[LT] J. Lindenstrauss and L. Tzarfriri, Classical Banach Spaces I, Springer-Verlag, Berlin, (1977).

[LW] R. Li and J. Wang, Invariants in Abstract Mappping Pairs, J. Aust. Math. Soc., 76, pp. 369-381, (2004).         [ Links ]

[MO] S. Mazur and W. Orlicz, Uber Folgen linearen Operationen, Studia Math., 4, pp. 152-157, (1933).         [ Links ]

[O] W. Orlicz, Beitrage zur Theorie der Orthogonalent Wichlungen II, Studia Math., 1, pp. 241-255, (1929).

[P] B. Pettis, On Integration in Vector Spaces, Trans. Amer. Math. Soc., 49, pp. 277-304, (1938).

[Pi] A. Pietsch, Nukleare Lokalconvexe Raume, Akademie Verlag, Berlin, (1965).         [ Links ]

[MO] A. Mohsen, Weak*-Norm Sequentially Continuous Operators, Math. Slovaca, 50, pp. 357-363, (2000).         [ Links ]

[R] A. Robertson, On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh, 68, pp. 145-157, (1969).

[Rol] S. Rolewicz, Metric Linear Spaces, Polish Sci. Publishers, Warsaw, (1984).

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

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

[Sw2] C. Swartz. A Generalized Orlicz-Pettis Theorem and Applications, Math. Z., 163, pp. 283-290, (1978).

[Sw3] C. Swartz, An Abstract Orlicz-Pettis Theorem, Bull. Pol. Acad. Sci. 32, pp. 433-437, (1984).         [ Links ]

[Sw4] C. Swartz, A Generalization of Stiles Orlicz-Pettis Theorem, Rend. Inst. Matem Univ. di Trieste, 20, pp. 109-112, (1988).

[Sw5] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ., Singapore, (1996).         [ Links ]

[Sw6] C. Swartz, Multiplier Convergent Series, World Sci. Publ., Singapore, (2009)        [ Links ]

[Sw7] C. Swartz. A Bilinear Orlicz-Pettis, Theorem, J. Math. Anal. Appl.,365, pp. 332-337, (2010).

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

[Tw] I. Tweddle, Unconditional Convergence and Vector-Valued Measures, J. London Math. Soc., 2, pp. 603-610, (1970).         [ Links ]

[We] H. Weber, Compactness in Spaces of Group-Valued Contents, the Vitali-Hahn-Saks Theorem and Nikodym Boundedness Theorem, Rocky. Mount. J. Math., 16, pp. 253-275, (1985).         [ Links ]

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

[ZCL] F. Zheng, C. Cui, and R. Li, Abstract Gliding Hump Properties in the Vector-Valued Dual Pairs, Acta Anal. Functionalis Appl., 12, pp. 322-327, (2010).         [ Links ]

Min-Hyung Cho

Department of Mathematics

Kum-Oh National Institute of Technology

Kumi, Korea

e-mail :

Li Ronglu

Department of Mathematics

Harbin Institute of Technology


P. R. C.

Charles Swartz

Department of Mathematics New Mexico State University

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

e-mail :

Received : October 2014. Accepted : October 2014

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