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Summability in L1 of a vector measure

We show a picture of the relations among different types of summability of series in the space L1(m) of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c0 in L1(m) , as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.