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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.

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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.

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"This book is a valuable contribution to the literature on Banach spaces." (David Yost, zbMATH 1379.46002, 2018)

"The authors provide an excellent presentation of the subject, and they manage to organize an impressive amount of material in such a way that, although they use a great variety of tools from various branches to prove the results, the work remains readable and thought-provoking. The book will be an indispensible resource for graduate students and researchers." (Antonis N. Manoussakis, Mathematical Reviews, January, 2017)

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