Covering a compact set in a Banach space by an operator range of a Banach space with basis
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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 p spaces and quotients of p spaces in terms of covering compact sets in X by operator ranges from p spaces. A Banach space X is a 1 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.