  We consider operators T : M_0 -> Z and T : M -> Z, where Z is a Banach space
and (M_0, M) is a pair of Banach spaces belonging to a general construction in
which M is defined by a "big-O" condition and M_0 is given by the corresponding
"little-o" condition. Prototype examples of such spaces M are given by
$\ell^\infty$, weighted spaces of functions or their derivatives, bounded mean
oscillation, Lipschitz-H\"older spaces, and many others. The main result
characterizes the weakly compact operators T in terms of a certain norm
naturally attached to M, weaker than the M-norm, and shows that weakly compact
operators T : M_0 -> Z are already quite close to being completely continuous.
Further, we develop a method to extract c_0-subsequences from sequences in M_0.
Applications are given to the characterizations of the weakly compact
composition and Volterra-type integral operators on weighted spaces of analytic
functions, BMOA, VMOA, and the Bloch space.
