Descriptive set theory and Banach spaces.

*(English)*Zbl 1121.46008
Johnson, W.B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1007-1069 (2003).

This handbook article consists of three, only loosely connected, parts.

Part I deals mainly with Baire-1 functions and pointwise compact sets of such functions on Polish spaces. Applications include Rosenthal’s \(\ell_1\)-theorem, a classification theorem for representable Banach spaces, and results concerning the complexity (in terms of descriptive set theory) of families of Banach spaces.

In Part II, Baire-1 elements of \(X^{**}\), considered as functions on the dual unit ball, are treated along with variants such as Baire-\(1/2\), Baire-\(1/4\) elements or differences of bounded semicontinuous functions. Containment of \(\ell_1\) or \(c_0\) in the Banach space \(X\) is expressed by means of these functions. The core of this part is a proof of Rosenthal’s \(c_0\)-theorem. A further subsection deals with spreading models.

Part III studies the complexity of weakly null sequences, a topic of many publications by the first named author. Special emphasis is put on various notions of restricted unconditionality and dichotomies of weakly null sequences. The final section deals with arbitrarily distortable and asymptotic \(\ell_p\)-spaces.

Each part is accompanied by a “Notes and remarks” section detailing the development of the main results and providing references to the original papers.

For the entire collection see [Zbl 1013.46001].

Part I deals mainly with Baire-1 functions and pointwise compact sets of such functions on Polish spaces. Applications include Rosenthal’s \(\ell_1\)-theorem, a classification theorem for representable Banach spaces, and results concerning the complexity (in terms of descriptive set theory) of families of Banach spaces.

In Part II, Baire-1 elements of \(X^{**}\), considered as functions on the dual unit ball, are treated along with variants such as Baire-\(1/2\), Baire-\(1/4\) elements or differences of bounded semicontinuous functions. Containment of \(\ell_1\) or \(c_0\) in the Banach space \(X\) is expressed by means of these functions. The core of this part is a proof of Rosenthal’s \(c_0\)-theorem. A further subsection deals with spreading models.

Part III studies the complexity of weakly null sequences, a topic of many publications by the first named author. Special emphasis is put on various notions of restricted unconditionality and dichotomies of weakly null sequences. The final section deals with arbitrarily distortable and asymptotic \(\ell_p\)-spaces.

Each part is accompanied by a “Notes and remarks” section detailing the development of the main results and providing references to the original papers.

For the entire collection see [Zbl 1013.46001].

Reviewer: Dirk Werner (Berlin)

##### MSC:

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

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

46B25 | Classical Banach spaces in the general theory |

54H05 | Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) |

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

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |