Subseries convergence in abstract duality pairs

Subseries convergence in abstract duality pairs

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}...

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Autores principales: Cho,Min-Hyung, Ronglu,Li, Swartz,Charles
Lenguaje:English
Publicado: Universidad Católica del Norte, Departamento de Matemáticas 2014
Acceso en línea:http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000400007
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spelling oai:scielo:S0716-091720140004000072015-01-19Subseries convergence in abstract duality pairsCho,Min-HyungRonglu,LiSwartz,CharlesLet 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 {y nk } such that limj; b(x, y nk) 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 {x nj} there is an element x G E such that Xj=! b(x nj ,y) = b(x,y) for every y G F ,then the series Xj=! b(x nj ,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.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.33 n.4 20142014-12-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000400007en10.4067/S0716-09172014000400007
institution Scielo Chile
collection Scielo Chile
language English
description 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 {y nk } such that limj; b(x, y nk) 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 {x nj} there is an element x G E such that Xj=! b(x nj ,y) = b(x,y) for every y G F ,then the series Xj=! b(x nj ,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 Cho,Min-Hyung
Ronglu,Li
Swartz,Charles
spellingShingle Cho,Min-Hyung
Ronglu,Li
Swartz,Charles
Subseries convergence in abstract duality pairs
author_facet Cho,Min-Hyung
Ronglu,Li
Swartz,Charles
author_sort Cho,Min-Hyung
title Subseries convergence in abstract duality pairs
title_short Subseries convergence in abstract duality pairs
title_full Subseries convergence in abstract duality pairs
title_fullStr Subseries convergence in abstract duality pairs
title_full_unstemmed Subseries convergence in abstract duality pairs
title_sort subseries convergence in abstract duality pairs
publisher Universidad Católica del Norte, Departamento de Matemáticas
publishDate 2014
url http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000400007
work_keys_str_mv AT chominhyung subseriesconvergenceinabstractdualitypairs
AT rongluli subseriesconvergenceinabstractdualitypairs
AT swartzcharles subseriesconvergenceinabstractdualitypairs
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