International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 37, Pages 2375-2378
doi:10.1155/S0161171203302315

DP1 and completely continuous operators

Elizabeth M. Bator1 and Dawn R. Slavens2

1Department of Mathematics, University of North Texas, P.O. Box 311400, Denton 76203-1400, TX, USA
2Department of Mathematics, Midwestern State University, 3410 Taft Blvd, Wichita Falls 76308, TX, USA

Received 11 February 2003

Copyright © 2003 Elizabeth M. Bator and Dawn R. Slavens. 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

W. Freedman introduced an alternate to the Dunford-Pettis property, called the DP1 property, in 1997. He showed that for 1≤p<∞, (⊕α∈𝒜Xα)p has the DP1 property if and only if each Xα does. This is not the case for (⊕α∈𝒜Xα)∞. In fact, we show that (⊕α∈𝒜Xα)∞ has the DP1 property if and only if it has the Dunford-Pettis property. A similar result also holds for vector-valued continuous function spaces.