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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.4 Antofagasta Dec. 2014 

Subseries convergence in abstract duality pairs

Min-Hyung Cho

Kum-Oh National Institute of Tech.


Li Ronglu

Harbin Institute of Tech.

P. R. C.

Charles Swartz

New Mexico State University



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.


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Min-Hyung Cho

Department of Mathematics

Kum-Oh National Institute of Technology

Kumi, Korea

e-mail :

Li Ronglu

Department of Mathematics

Harbin Institute of Technology


P. R. C.

Charles Swartz

Department of Mathematics New Mexico State University

Las Cruces, NM, 88003, U. S. A.

e-mail :

Received : October 2014. Accepted : October 2014

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