We consider a general framework of Banach spaces M defined by a boundedness condition. Prototype examples are l^infty, weighted spaces of functions or their derivatives, BMO, Lipschitz spaces, and many others. Each space M has a small space counterpart M_0, consisting of elements of M where the quantity of the boundedness condition "vanishes at infinity". In the above examples, the corresponding M_0-spaces are c_0, vanishing weighted spaces, VMO, small Lipschitz spaces, etc. We will show that the bidual of M_0 is always M, as expected. Furthermore, letting T be a continuous operator T : M_0 -> Z, Z any Banach space, it turns out that if T is weakly compact, it is already very close to being compact. This is remarkable since weak compactness is in general a very weak property. The phenomenon has been observed previously for classes of concrete operators such as composition operators and integration operators acting on spaces of analytic functions with supremum-type norms, where it often happens that weak compactness and compactness coincide.