  Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y'
--> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y'
respectively. We obtain apparently new estimates for the covering numbers of
the set S(B(Y')). These are expressed in terms of the covering numbers of
T(B(X)), or, more generally, in terms of the covering numbers of a
"significant" subset of T(B(X)). The latter more general estimates are best
possible. These estimates follow from our new quantitative version of an
abstract compactness result which generalizes classical theorems of
Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering
numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of
a suitable "significant" subset of S(B(Y')).
