Nonseparable Banach spaces.

*(English)*Zbl 1041.46009
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1743-1816 (2003).

At the core of this survey are three groups of properties of a (nonseparable) Banach space \(X\):

1. Properties of the norm such as fragmentability, and massiveness of the set of smooth points and extreme points. The possibility of introducing equivalent norms with different kinds of convexity and smoothness.

2. The existence of sufficiently many projections (projectional resolutions), and related topics such as various types of Markushevich basis and quasicomplements.

3. Sequential properties of the weak and weak* topologies.

These three groups of properties are tightly interlaced: projectional resolutions allow, using transfinite induction, the construction of better equivalent norms; nice norms give nice properties of the weak topologies; good sequential properties of the ball \(B(X^*)\) permit the construction of projectional resolutions and so on. With this point of view, the author has not only described a large body of facts, but also organized them well. Crucial for him are equivalent norms: a half of the survey and the majority of unsolved problems are devoted to them.

The main types of nonseparable Banach spaces such as weakly compactly generated, weakly Lindelöf and others are described in detail. Also described are important examples which distinguish between different properties. The tight connection between nonseparable Banach spaces and nonmetrizable compact spaces is indicated: properties of the Banach space \(C(K)\) are tightly connected with properties of the compactum \(K\) and properties of a general Banach space \(X\) with properties of the weak* compactum \(B(X^*)\).

Of course, nonseparable Banach space theory is not restricted to the above mentioned topics. Further topics can be found by the reader in other articles of this Handbook, and in the wide-ranging bibliography of this survey which contains 323 items. So, the paper under review is itself a useful handbook for these topics; the extensive material is not simply collected, but the connections, motivations, history and perspectives are clearly indicated.

For the entire collection see [Zbl 1013.46001].

1. Properties of the norm such as fragmentability, and massiveness of the set of smooth points and extreme points. The possibility of introducing equivalent norms with different kinds of convexity and smoothness.

2. The existence of sufficiently many projections (projectional resolutions), and related topics such as various types of Markushevich basis and quasicomplements.

3. Sequential properties of the weak and weak* topologies.

These three groups of properties are tightly interlaced: projectional resolutions allow, using transfinite induction, the construction of better equivalent norms; nice norms give nice properties of the weak topologies; good sequential properties of the ball \(B(X^*)\) permit the construction of projectional resolutions and so on. With this point of view, the author has not only described a large body of facts, but also organized them well. Crucial for him are equivalent norms: a half of the survey and the majority of unsolved problems are devoted to them.

The main types of nonseparable Banach spaces such as weakly compactly generated, weakly Lindelöf and others are described in detail. Also described are important examples which distinguish between different properties. The tight connection between nonseparable Banach spaces and nonmetrizable compact spaces is indicated: properties of the Banach space \(C(K)\) are tightly connected with properties of the compactum \(K\) and properties of a general Banach space \(X\) with properties of the weak* compactum \(B(X^*)\).

Of course, nonseparable Banach space theory is not restricted to the above mentioned topics. Further topics can be found by the reader in other articles of this Handbook, and in the wide-ranging bibliography of this survey which contains 323 items. So, the paper under review is itself a useful handbook for these topics; the extensive material is not simply collected, but the connections, motivations, history and perspectives are clearly indicated.

For the entire collection see [Zbl 1013.46001].

Reviewer: Anatolij Plichko (Krakow)

##### MSC:

46B26 | Nonseparable Banach spaces |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B10 | Duality and reflexivity in normed linear and Banach spaces |

54D30 | Compactness |

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