Some new rich subspaces of C. Applications

AbstractIn this paper, we are interested in a class of subspaces of C, introduced by Bourgain [Studia Math. 77 (1984) 245–253]. Wojtaszczyk called them rich in his monograph [Banach Spaces for Analysts, Cambridge Univ. Press, 1991]. We give some new examples of such spaces: this allows us to recover previous results of Godefroy–Saab and Kysliakov on spaces with reflexive annihilator in a very simple way. We construct some other examples of rich spaces, hence having property (V) of Pełczyński and Dunford–Pettis property. We also recover the results due to Bourgain and Saccone saying that spaces of uniformly convergent Fourier series share these properties, by only using the main result of [Studia Math. 77 (1984) 245–253] and some very elementary arguments. We generalize too these results