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**Asplund spaces for beginners.**
*(English)*
Zbl 0815.46022

This is a beautiful survey on Asplund spaces, which are a special kind of Banach space: those having dual with the Radon-Nikodym, or the Krein- Milman property; those for which every separable subspace has a separable dual …They, moreover, enjoy many other interesting properties, such as being dual ball weak* sequentially compact.

The author starts from scratch and takes the reader for a tournée through each delicate and/or interesting point in the theory without even a tilt of the hat. Indeed, not only complete proofs are given for almost every result, no, that would be too easy! When the equivalence of several formulations for Asplundness is stated then the author shows the way to go to each of them starting from any other. This is what “explain” means for those who think that in mathematics not only what is done counts, but also how is it done. This paper could be put as the perfect example of what a survey sould be.

I am not going to make any further attempt to explain what Asplund spaces are or what they are for. If you want to know anything about them, read this survey.

The author starts from scratch and takes the reader for a tournée through each delicate and/or interesting point in the theory without even a tilt of the hat. Indeed, not only complete proofs are given for almost every result, no, that would be too easy! When the equivalence of several formulations for Asplundness is stated then the author shows the way to go to each of them starting from any other. This is what “explain” means for those who think that in mathematics not only what is done counts, but also how is it done. This paper could be put as the perfect example of what a survey sould be.

I am not going to make any further attempt to explain what Asplund spaces are or what they are for. If you want to know anything about them, read this survey.

Reviewer: J.M.F.Castillo (Badasoz)

### MSC:

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

46B20 | Geometry and structure of normed linear spaces |

46G05 | Derivatives of functions in infinite-dimensional spaces |