Extremely weakly unconditionally convergent series

It is proved that if Σ _i=1 ^∞ X _i is a non-convergent series in a Banach spaceX such that Σ _i=1 ^∞ |f(X _i )|<∞ for all extreme pointsf of the unit ball ofX*, thenX contains a subspace isomorphic toc ₀, improving a result of Bessaga and Pelczynski. The proof uses Fonf’s result that Lindenstrauss-Phelps spaces contain isomorphs ofc ₀. Supported in part by NSF-MCS-8002393.