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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.
This book on Banach space theory focuses on what have been called
three-space problems. It contains a fairly complete description of
ideas, methods, results and counterexamples. It can be considered
self-contained, beyond a course in functional analysis and some
familiarity with modern Banach space methods. It will be of
interest to researchers for its methods and open problems, and to
students for the exposition of techniques and examples.
Many researchers in geometric functional analysis are unaware of
algebraic aspects of the subject and the advances they have
permitted in the last half century. This book, written by two world
experts on homological methods in Banach space theory, gives
functional analysts a new perspective on their field and new tools
to tackle its problems. All techniques and constructions from
homological algebra and category theory are introduced from scratch
and illustrated with concrete examples at varying levels of
sophistication. These techniques are then used to present both
important classical results and powerful advances from recent
years. Finally, the authors apply them to solve many old and new
problems in the theory of (quasi-) Banach spaces and outline new
lines of research. Containing a lot of material unavailable
elsewhere in the literature, this book is the definitive resource
for functional analysts who want to know what homological algebra
can do for them.
This book presents an overview of modern Banach space theory. It
contains sixteen papers that reflect the wide expanse of the
subject. Articles are gathered into five sections according to
methodology rather than the topics considered. The sections are:
geometrical methods; homological methods; topological methods;
operator theoretic methods; and also function space methods. Each
section contains survey and research papers describing the
state-of-the-art in the topic considered as well as some of the
latest most important results. Researchers working in Banach space
theory, functional analysis or operator theory will find much of
interest here.
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