On classes of Banach spaces admitting "small" universal spaces

الأصل · EN

We characterize those classes of separable Banach spaces admitting a separable universal space Y (that is, a space Y containing, up to isomorphism, all members of) 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 ∞-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, ∞-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.