Abstract: 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 (universally) separably injective spaces, including ultraproducts built over countably incomplete ultrafilters, in spite of the fact that these ultraproducts are never injective. We obtain two fundamental characterizations of universally separably injective spaces. (a) A Banach space is universally separably injective if and only if every separable subspace is contained in a copy of inside . (b) A Banach space is universally separably injective if and only if for every separable space one has . Section 6 focuses on special properties of 1-separably injective spaces. Lindenstrauss proved in the middle sixties that, under CH, 1-separably injective spaces are 1-universally separably injective and left open the question in ZFC. We construct a consistent example of a Banach space of type which is 1-separably injective but not universally 1-separably injective.

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