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A nonlinear transfer technique for renorming. (English) Zbl 1182.46001
Lecture Notes in Mathematics 1951. Berlin: Springer (ISBN 978-3-540-85030-4/pbk). xi, 142 p. (2009).
Renorming a Banach space means replacing its original norm, which is frequently provided by the very definition of the space, by an equivalent one which has better (or sometimes, but rarely, worse) properties of convexity and smoothness. It is frequently so that one tries to construct a norm which shares some properties with Euclidean norms, since the Hilbert space is somehow the “best” Banach space. Many examples show that the existence of a norm with specific properties of convexity and/or smoothness says a lot on the isomorphic properties of the relevant space.
It is by now well-known that Banach spaces usually fail to have a proper system of coordinates, and as a result computing in arbitrary Banach spaces can be difficult. This is especially true in non-separable Banach spaces, whose renormings are particularly challenging.
The Spanish school, and in particular the authors of the book under review, contributed a lot to the progress achieved in renorming theory over the last fifteen years. The present work gathers most of their contributions, from a unified and renewed point of view. Therefore, this book should be understood as an original publication rather than a survey.
Technicalities are quite necessary in renorming theory and this is just a fact of life. I should however, underline the fact that a successful effort has been made throughout the book for explaining in simple terms what goes on. Very roughly, one may describe the gist of this work as follows. If a Banach space \(Y\) has a strictly convex norm and there is a one-to-one continuous linear map \(T\) from a Banach space \(X\) to \(Y\), then \(X\) has an equivalent strictly convex norm. This is very simple, and it is also simple to check that it fails to work when we wish to carry stronger properties such as local uniform rotundity (LUR). Nonetheless, the linear transfer method asserts, for instance, that, if the biconjugate operator \(T^{**}\) is one-to-one and there is a LUR norm on \(Y\), then there is a LUR norm on \(X\). This last assumption can be shown to imply that the map \(T\) is \(\sigma\)-co-continuous, which somehow means that its inverse \(T^{-1}\) has similar properties to a first Baire class function. But once this point of view is reached, it is natural to wonder whether one can dispense with linearity of the \(\sigma\)-co-continuous map \(T\). The answer to this query is essentially positive, as shown in this book. It allows, in particular, to unify results which were shown through the transfer technique with those which followed from three-space arguments, and actually to prove almost all LUR renorming theorems known to this day. I should say that these “fragmentation” techniques have proven to be spectacularly useful in renorming theory: for instance, the theorem (due to the authors of this book) according to which every weakly LUR-renormable space is actually LUR renormable, and the more recent theorem (due to Richard Haydon), asserting that the existence of a dual LUR norm on \(X^*\) implies the existence of a LUR norm on \(X\), are spectacular achievements. It was, in fact, quite unexpected that the existence of such norms would provide enough information for reaching such a general conclusion.
This nice and deep book addresses problems which are of interest for every functional analyst, and, moreover, anyone who intends to contribute to renorming theory must read it. I should finally mention that a very interesting list of commented problems concludes the book. This list constitutes an attractive research program, which should stimulate research for the years to come.

MSC:
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
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