We show that ifXis a tight subspace ofC(K) thenXhas the PelŁczyński property andX* is weakly sequentially complete. We apply this result to the spaceUof uniformly convergent Taylor series on the unit circle and using a minimal amount of Fourier theory prove thatUhas the PelŁczyński property andU* is weakly sequentially complete. Using separate methods, we proveUandU* have the Dunford-Pettis property. Some results concerning pointwise bounded approximation are proved for tight uniform algebras. We use tightness and the PelŁczyński property to make a remark about inner functions on strictly pseudoconvex domains in Cn.