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Subseries convergence in abstract duality pairs

  • Cho, Min Hyung [3] ; Ronglu, Li [1] ; Swartz, Charles [2]
    1. [1] Harbin Institute of Technology

      Harbin Institute of Technology

      China

    2. [2] New Mexico State University

      New Mexico State University

      Estados Unidos

    3. [3] Kum-Oh National Institute of Tech.
    Mostrar afiliaciones +
  • Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 33, Nº. 4, 2014, págs. 447-470
  • Idioma: inglés
  • DOI: 10.4067/S0716-09172014000400007
  • Enlaces
  • Resumen

    • 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.

  • Referencias bibliográficas
    • Citas [AB] P. Antosik, Z. Bu, R. Li, E. Pap and C. Swartz, K -convergent sequences in locally convex spaces, Generalized Functions and Convergence,...
    • [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,...
    • [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.,‏...
    • [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....
    • [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.‏...
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