The Pelczynski property for tight subspaces

Abstract

We show that if X is a tight subspace of C(K) then X has the Pelczynski property and X^* is weakly sequentially complete. We apply this result to the space U of uniformly convergent Taylor series on the unit circle and using a minimal amount of Fourier theory prove a theorem of Bourgain, namely that U has the Pelczynski property and U^* is weakly sequentially complete. Using separate methods, we prove U and U^* have the Dunford-Pettis property. Some results concerning pointwise bounded approximation are proved for tight uniform algebras. We use tightness and the Pelczynski property sto make a remark about inner functions on strictly pseudoconvex domains in C^n.