Subseries convergence in abstract duality pairs
DOI:
https://doi.org/10.4067/S0716-09172014000400007Keywords:
Convergence series, vectorial spaces, Orlicz-Pettis theorem, abstract triples, series convergentes, espacios vectoriales, teorema de Orlicz-Pettis, triples abstractos.Abstract
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.References
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[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).
[AS1] P. Antosik and C. Swartz, Matrix Methods in Analysis, SingerVerlag, Heidelberg, (1985).
[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 OrliczPettis Theorem, Rend. Inst. Matem. Univ. di Trieste, 18 (1986), pp. 159-162, (1986).
[BK] G. Bennett and N. Kalton, FK-spaces containing c0, Duke Math. J., 39, pp. 561-582, (1972).
[C] C. Constantinescu, On Nikodyms Boundedness Theorem, Libertas Math., 1, pp. 51-73, (1981).
[BCS] O. Blasco, J. M. Calabuig and T. Signes, A bilinear version of OrliczPettis Theorem, J. Math. Anal. Appl., 348, pp. 150-164, (2008).
[CL] A. Chen and R. Li, A version of Orlicz-Pettis Theorem for quasihomogeneous operator space, J. Math. Anal. Appl., 373, pp. 127-133, (2011).
[Diel] P. Dierolf, Theorems of Orlicz-Pettis type for locally convex spaces, Man. Math., 20, pp. 73-94, (1977).
[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).
[Di] N. Dinculeanu, Weak Compactness and Uniform Convergence of Operators in Spaces of Bochner Integrable Function, J. Math. Anal. Appl., 109 (1985).
[Dr] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems, Bull. Aced. Polon. Sci., 20, pp. 725-731, (1972).
[DS] N. Dunford and J. Schwartz, Linear Operators, Interscience, N.Y., (1958).
[G] H. Garnir, M. DeWilde and J. Schmets, Analyse Fontionnelle I, Birkhauser, Basel, (1968).
[FL] W. Filter and I. Labuda, Essays on the Orlicz Theorem I, Real Anal. Exch., 16, pp. 393-403, (1990-1991).
[K2] N. Kalton, Subseries Convergence in Topological Groups and Vector Spaces, Israel J. Math., 10, pp. 402-412, (1971).
[K3] N. Kalton, Spaces of Compact Operators, Math. Ann., 208, pp. 267-278, (1974).
[Kh] S. M. Khaleelulla, Counter Examples in Topological Vector Spaces, Springer Lecture Notes in Mathematics 936, Heidelberg, (1982).
[LC] R.Li and M. Cho, A General Kalton-type Theorem, J. Harbin Inst. Tech 25, pp. 100-104, (1992).
[LS1] R. Li and C. Swartz, Spaces for which the Uniform Boundedness Principle Holds, Studia Sci. Math. Hungar., 27, pp. 379-384, (1992).
[LS2] R. Li and C. Swartz, A Nonlinear Schur Theorem, Acta Sci. Math. (Szged), 58, pp. 497-508, (1993).
[LS3] R. Li and C. Swartz, An Abstract Orlicz-Pettis Theorem and Applications, Proy. J. Math., 27, pp. 155-169, (2008).
[LT] J. Lindenstrauss and L. Tzarfriri, Classical Banach Spaces I, SpringerVerlag, Berlin, (1977).
[LW] R. Li and J. Wang, Invariants in Abstract Mappping Pairs, J. Aust. Math. Soc., 76, pp. 369-381, (2004).
[MO] S. Mazur and W. Orlicz, Uber Folgen linearen Operationen, Studia Math., 4, pp. 152-157, (1933).
[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).
[MO] A. Mohsen, Weak*-Norm Sequentially Continuous Operators, Math. Slovaca, 50, pp. 357-363, (2000).
[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).
[Sw1] C. Swartz, An Introduction to Functional Analysis, Dekker, N.Y., (1992).
[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).
[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).
[Sw6] C. Swartz, Multiplier Convergent Series, World Sci. Publ., Singapore, (2009).
[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).
[Tw] I. Tweddle, Unconditional Convergence and Vector-Valued Measures, J. London Math. Soc., 2, pp. 603-610, (1970).
[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).
[Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, Mc Graw-Hill, N. Y. (1978).
[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).
Published
2017-03-23
How to Cite
[1]
M. H. Cho, L. Ronglu, and C. Swartz, “Subseries convergence in abstract duality pairs”, Proyecciones (Antofagasta, On line), vol. 33, no. 4, pp. 447-470, Mar. 2017.
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