A note on p-limited sets

Given p⩾1p⩾1, a subset A of a Banach space X is said to be p-limited if for every weakly p  -summable sequence (xn⁎) in X⁎X⁎ there exists (αn)∈ℓp(αn)∈ℓp such that |〈xn⁎,x〉|⩽αn for all x∈Ax∈A and n∈Nn∈N. It is showed that p-limited sets are q  -limited whenever p