A note on p-limited sets

Abstract

Given p⩾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⁎ there exists (αn)∈ℓp such that |〈xn⁎,x〉|⩽αn for all x∈A and n∈N. It is showed that p-limited sets are q-limited whenever p<q and Banach spaces enjoying the property that every q-limited subset is p-limited are characterized. We also prove that an operator has p-summing adjoint if and only if it maps relatively compact sets to p-limited sets.