ANGELICITY AND THE BOUNDARY-PROBLEM

Let K be an arbitrary compact space and C(K) the space of continuous f unctions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K) on which every function of C (K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology t(p)(B) of pointwise conv ergence on B. It is also shown that this result can be extended to a l arge class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), t(p)(B)) is an ange lic space in the sense of D. H. Fremlin.