Subseries convergence in abstract duality pairs


  • Min Hyung Cho Kum-Oh National Institute of Tech.
  • Li Ronglu Harbin Institute of Technology.
  • Charles Swartz New Mexico State University.



Convergence series, vectorial spaces, Orlicz-Pettis theorem, abstract triples, series convergentes, espacios vectoriales, teorema de Orlicz-Pettis, triples abstractos.


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, ynkexists 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.

Author Biographies

Min Hyung Cho, Kum-Oh National Institute of Tech.

Department of Mathematics.

Li Ronglu, Harbin Institute of Technology.

Department of Mathematics.

Charles Swartz, New Mexico State University.

Department of Mathematics.


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How to Cite

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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