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: | , , |
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| Lenguaje: | English |
| Publicado: |
Universidad Católica del Norte, Departamento de Matemáticas
2014
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| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172014000400007 |
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| Sumario: | 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. |
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