On Classes of Banach Spaces Admitting “small” Universal Spaces

We characterize those classes C of separable Banach spaces admitting a separable universal space Y (that is, a space Y containing, up to isomorphism, all members of C) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class NU of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form [AD]. Our approach is based, among others, on a construction of L∞-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family {Yξ : ξ < ω1} of separable, non-universal, L∞-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.