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.

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• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 33, Nº. 4, 2014, págs. 447-470
• Idioma: inglés
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• 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.