Separably Injective Banach Spaces

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we present the basic characterizations and a number of structural properties of (universally) separable injective Banach spaces. We will show, among other things, that 1-separably injective spaces are not necessarily isometric to C-spaces, that (universally) separably injective spaces are not necessarily complemented in any C-space—the separably injective part of the assertion will be shown here while the “universal” part can be found in the next chapter—and that there exist essential differences between 1-separably injective and 2-separably injective spaces. Moreover, in contrast with the scarcity of examples and general results concerning the class of injective Banach spaces, there exist many different types of separably injective spaces and a rich theory around them. In fact, most of the chapter is devoted to examples: Some of them are rather natural, while others are Banach spaces introduced elsewhere for different purposes and that, at the end of the day, turn out to be separable injective.