2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169647For every $α<ω_1$ we establish the existence of a separable Banach space whose Szlenk index is $ω^{αω+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $ω^{αω}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.Functional AnalysisAlgebraic Geometry46B20, 54H05text