Covering a compact set in a banach space by an operator range of a banach space with basis

A Banach space X has the approximation property if and only if every compact set in X is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for ℒ_p spaces and quotients of ℒ_p spaces in terms of covering compact sets in X by operator ranges from ℒ_p spaces. A Banach space X is a ℒ₁ space if and only if every compact set in X is contained in the closed convex symmetric hull of a basic sequence which converges to zero.