Maps with Separably Injective Banach Spaces

Abstract

We show maps whose values are on Banach Space. We also show a method to obtain Banach spaces of universal and almost -universal disposition with respect to a given class of normed spaces. The method produces, among other, the Gurariĭ space or kubis space .We deal with two weak forms of injectivity which turn out to have a rich structure behind: separable injectivity and universal separable injectivity. We show several structural and stability properties of these classes of Banach spaces. We provide natural examples of separably injective spaces, including l_∞ ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We also find a nonseparable generalization which is of universal disposition for separable spaces and “separably injective”. No separably injective p-Banach space was previously known for p<1.