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.