Randomized series and geometry of Banach spaces

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that lⁿ_∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner L_p(X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is uniformly monotone if and only if its p-convexification E^(p) is uniformly convex and that a Köthe function space E is upper locally uniformly monotone if and only if its p-convexification E^(p) is midpoint locally uniformly convex.