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Journal of Function Spaces
Volume 2016, Article ID 3763649, 8 pages
http://dx.doi.org/10.1155/2016/3763649
Research Article

Topological Dual Systems for Spaces of Vector Measure -Integrable Functions

1Departamento de Análisis Matemático, Universidad de Valencia, Burjassot, 46100 Valencia, Spain
2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain

Received 21 April 2016; Accepted 30 May 2016

Academic Editor: Miguel Martín

Copyright © 2016 P. Rueda and E. A. Sánchez Pérez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show a Dvoretzky-Rogers type theorem for the adapted version of the -summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.