Abstract.Among Bourgain's many remarkable theorems is one from 1980 which states that if T is a non-Dunford-Pettis operator from L| [0,1] The first section collects terminology, gives the introductory results and shows that the NRNP is a three space property. The second section studies a continuity property that implies near representability. Vf(t)= [ fdp, 0 < r < 1.
JoProbably the most intriguing fact about this operator is that it is completely continuous, i.e., it sends weakly compact (uniformly integrable) sets in L, [0,1] into compact (equicontinuous) sets in C [0,1]. Once this fact is realized, it is
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