A quantitative extension of a theorem of Valdivia on weak compactness

We prove the following result: Let M be a subset of a Banach space X with the following property: there is ε≥0such that f is bounded on M whenever f is a real-valued and σ(X″,X′)-continuous function on X+εBX″. Then M is 2ε-weakly relatively compact, that is, the σ(X″,X′)-closure of M is contained in X+2εBX″. This theorem is a quantitative extension of a theorem on weak compactness due to M. Valdivia that corresponds to the case ε=0.