DP1 and completely continuous operators - Elizabeth M. Bator; Dawn R. Slavens (International Journal of Mathematics and Mathematical Sciences)
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.