Banach spaces and topology. II.

*(English)*Zbl 0832.46005
Hušek, Miroslav (ed.) et al., Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 493-536 (1992).

This article which deals with some aspects of topological methods in Banach spaces is a continuation of the survey article “Banach Spaces and Topology” that appeared in the Handbook of Set-Theoretic Topology, cf. part I, the second author, 1045-1142 (1984; Zbl 0584.46007). We rely on this survey article for notation and terminology. For many results only statements, or rough indications of their proofs are given. Some open questions are stated.

We start, in Section 2, with the principal results on Namioka’s properties on jointly continuous functions. The results here have a strong topological flavor, however their motivation to a large extent springs from renormings of Banach spaces, with norms having mainly desirable convexity type properties. [For such problems the reader is referred to the authoritative text of R. Deville, G. Godefroy and V. Zizler, ‘Smoothness and renormings in Banach spaces’ (1993; Zbl 0782.46019).]

In Section 3, the complexity of Baire-1 functions, as described by a countable ordinal index, produces some local analogues (spreading models) of Rosenthal’s fundamental theorem on \(\ell^1\)-embeddings [cf. part I, loc. cit., Section 1].

In Section 4, we study the class of weakly Lindelöf determined (WLD) Banach spaces (coinciding with those Banach spaces which have a Corson compact dual unit ball in its weak\(^*\) topology). The original method of projections, [described in part I, Section 6] is extended to the largest, in a sense, class of spaces within which many of the original basic characteristics of the method are preserved.

In Sections 5 and 6 we consider renorming problems for the class of WLD Banach spaces and we clarify, with examples, the significant difference of the behavior of this class relative to smaller classes such as the countably determined Banach spaces.

Section 7 is concerned with the method of projections as it is applied to the class of dual Banach spaces with Radon-Nikodým property (Fabian- Godefroy theorem).

In Section 8, we study the fragmented and the Radon-Nikodým compact spaces, and in particular the Orihuela-Schachermayer-Valdivia and Stegall result (to the effect that every Corson compact Radon-Nikodým compact space is necessarily Eberlein compact).

For the entire collection see [Zbl 0782.00072].

We start, in Section 2, with the principal results on Namioka’s properties on jointly continuous functions. The results here have a strong topological flavor, however their motivation to a large extent springs from renormings of Banach spaces, with norms having mainly desirable convexity type properties. [For such problems the reader is referred to the authoritative text of R. Deville, G. Godefroy and V. Zizler, ‘Smoothness and renormings in Banach spaces’ (1993; Zbl 0782.46019).]

In Section 3, the complexity of Baire-1 functions, as described by a countable ordinal index, produces some local analogues (spreading models) of Rosenthal’s fundamental theorem on \(\ell^1\)-embeddings [cf. part I, loc. cit., Section 1].

In Section 4, we study the class of weakly Lindelöf determined (WLD) Banach spaces (coinciding with those Banach spaces which have a Corson compact dual unit ball in its weak\(^*\) topology). The original method of projections, [described in part I, Section 6] is extended to the largest, in a sense, class of spaces within which many of the original basic characteristics of the method are preserved.

In Sections 5 and 6 we consider renorming problems for the class of WLD Banach spaces and we clarify, with examples, the significant difference of the behavior of this class relative to smaller classes such as the countably determined Banach spaces.

Section 7 is concerned with the method of projections as it is applied to the class of dual Banach spaces with Radon-Nikodým property (Fabian- Godefroy theorem).

In Section 8, we study the fragmented and the Radon-Nikodým compact spaces, and in particular the Orihuela-Schachermayer-Valdivia and Stegall result (to the effect that every Corson compact Radon-Nikodým compact space is necessarily Eberlein compact).

For the entire collection see [Zbl 0782.00072].

##### MSC:

46Bxx | Normed linear spaces and Banach spaces; Banach lattices |

46A50 | Compactness in topological linear spaces; angelic spaces, etc. |

54D30 | Compactness |

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

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