This monograph contains a detailed exposition of the up-to-date theory of separably injective spaces: new and old results are put into perspective with concrete examples (such as l/c0 and C(K) spaces, where K is a finite height compact space or an F-space, ultrapowers of L spaces and spaces of universal disposition).

It is no exaggeration to say that the theory of separably injective Banach spaces is strikingly different from that of injective spaces. For instance, separably injective Banach spaces are not necessarily isometric to, or complemented subspaces of, spaces of continuous functions on a compact space. Moreover, in contrast to the scarcity of examples and general results concerning injective spaces, we know of many different types of separably injective spaces and there is a rich theory around them. The monograph is completed with a preparatory chapter on injective spaces, a chapter on higher cardinal versions of separable injectivity and a lively discussion of open problems and further lines of research.

This is the first book dedicated to the subject Most of the material has never appeared in book form before (and all references to external results have been restricted to material which can be found in books) The exposition is lively and detailed, in contrast to the typical style of mathematical papers The monograph contains many annotated open problems, alternative routes and suggestions for new lines of research. Includes supplementary material: sn.pub/extras

Inhalt
A primer on injective Banach spaces.- Separably injective Banach spaces.- Spaces of universal disposition.- Ultraproducts of type L .- -injectivity.- Other weaker forms of injectivity.- Open Problems.