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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

CHO, Min-Hyung; RONGLU, Li and SWARTZ, Charles. **Subseries convergence in abstract duality pairs**.* Proyecciones (Antofagasta)* [online]. 2014, vol.33, n.4, pp.447-470.
ISSN 0716-0917. http://dx.doi.org/10.4067/S0716-09172014000400007.

*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*Xj=! b(x

_{nj}} there is an element x G E such that_{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.*