An Asplund space with norming Markuševič basis that is not weakly compactly generated☆
Introduction
The crystallisation, in the mid-sixties, of the notions of projectional resolution of the identity (PRI, for short) [58] and of weakly compactly generated Banach space (WCG) [5] opened the way to a spectacular development in Banach space theory, leading to a structural theory for many classes of non-separable Banach spaces. Just to mention some advances, we refer, e.g., to [8], [21], [36], [48], [69], [75], [83]. Such a theory is tightly connected to differentiability [12], [30], [31], [33], [34], [45], classes of compacta [10], [15], [16], [19], [20], [35], [52], [59], combinatorics [6], [25], [26], [62], [66], [72], [80], [78].
An important tool in the area was introduced by Fabian [27], who used Jayne–Rogers selectors [43] to show that every weakly countably determined Asplund Banach space is indeed WCG. Jayne–Rogers selectors were also deeply involved, together with Simons' lemma [73], [38], in the proof that the dual of every Asplund space admits a PRI, [29]. The techniques of [27] also used ingredients from [45], where it is shown, among others, that WCG Banach spaces with a Fréchet smooth norm admit a shrinking M-basis. Results of this nature led to the conjecture that Asplund Banach spaces with a norming M-basis are WCG. This question is originally due to Godefroy, who, at the times when [24] was in preparation, conjectured that a similar use of Jayne–Rogers selectors might produce a linearly dense weakly compact subset, in presence of a norming M-basis.
Problem 1.1 G. Godefroy Let be an Asplund space with a norming Markuševič basis. Must be weakly compactly generated?
Theorem A There exists an Asplund space with a 1-norming M-basis such that is not WCG.
Since the result [5] that WCG Banach spaces admit an M-basis, and, therefore, reflexive spaces have a shrinking basis, it readily became clear that M-bases with additional properties would have been instrumental in the characterisation of several classes of Banach spaces, [41, Chapter 6], [83]. In particular, it was natural to ask which class of Banach spaces is characterised by the presence of a norming M-basis. This led to the famous question, due to John and Zizler, whether every WCG Banach space admits a norming M-basis [46], that was recently solved in the negative by the first-named author, [40]. In this sense, Problem 1.1 can be considered as a converse to the said John's and Zizler's question.
As it turns out, there is an elegant characterisation of Banach spaces that admit a shrinking M-basis, in the form of the following result, due to the efforts of many mathematicians, [27], [81], [65], [44], [45]. We refer to [41, Theorem 6.3], or [28, Theorem 8.3.3] for a proof.
Theorem 1.2 For a Banach space , the following are equivalent: admits a shrinking M-basis; is WCG and Asplund; is WLD and Asplund; is WLD and has a dual LUR norm; is WLD and it admits a Fréchet smooth norm.
Problem 1.1 is clearly closely related to this result, since it amounts to asking whether the assumption in (ii) that is WCG could be replaced by the existence of a norming M-basis.
Our construction in Theorem A heavily depends on the existence of a peculiar scattered compact space, whose properties we shall record in Theorem B below. Before its statement, we need one piece of notation.
Given a set S, we identify the power set with the product , via the canonical correspondence (). Since is a compact topological space in its natural product topology, this identification allows us to introduce a compact topology on . Throughout our article, any topological consideration relative to will refer to the said topology, that we shall refer to as the product, or pointwise, topology.
Theorem B There exists a family of finite subsets of such that has the following properties: for every , for every , if is an infinite set, then for some , is scattered.
The subscript ϱ in our notation for the family reflects the rôle of the choice of a ϱ-function, a rather canonical semi-distance on , in the construction of the family . ϱ-functions were introduced in [77] for a study of the way Ramsey's theorem fails in the uncountable context. They appeared already in Banach space constructions, see, e.g., [7], [13], [61], [62]. We refer to [79], [11], [14] for a detailed presentation of this theory and further applications in several areas. ϱ-functions are also tightly related to construction schemes, [80], that also proved very useful in non-separable Banach space theory, see, e.g., [60].
In conclusion to this section, we briefly describe the organisation of the paper. Section 2 contains a revision of the notions from non-separable Banach space theory that are relevant to our paper. The proof of Theorem B, together with a quick revision of the necessary results concerning ϱ-functions, will be given in Section 3. Section 4 is independent from the argument in Section 3, as it only depends on the statement of Theorem B. Apart for the proof of Theorem A, we observe there that the compact space in Theorem B also offers an interesting example for the theory of semi-Eberlein compacta, solving a problem from [55]. Finally, in Section 5, we discuss the main problem in the case, where is an adequate compact; in particular, we show that has a 1-norming M-basis, whenever is adequate. Finally, a typographical note: the symbol ■ denotes the end of a proof, while, in nested proofs, we use □ for the end of the inner proof.
Section snippets
General conventions
Our notation concerning Banach spaces is standard, as in most textbooks in Banach space theory; we refer, e.g., to [1], [32]. All our results in the paper are valid for Banach spaces over either the real or the complex fields, with the same proofs. By a subspace of a Banach space we understand a closed, linear subspace.
We indicate by the cardinality of a set S. For a set S and a cardinal number κ, we write and . We denote by ω the first infinite ordinal and
The proof of Theorem B
The goal of the present section is the proof of Theorem B. As we already mentioned in the Introduction, the construction of the family depends upon the choice of a ϱ-function with some additional properties. Therefore, we start the section recalling some facts concerning ϱ-functions.
We will consider functions and it will be convenient to identify their domain with the set This just amounts to replacing the unordered pair with the ordered one , where
Proof of Theorem A
This section is dedicated to the proof of our main result.
Proof of Theorem A According to Theorem B, we can pick a family such that the compact has the following properties: for every , for every , if is an infinite set, then for some , is scattered.
We define a biorthogonal system in the Banach space as follows. For , let Note that, by (i), for each
Adequate compacta and norming M-bases
In conclusion to our paper, we shall briefly discuss the main problem in the case of a space, where is an adequate compact. We note the easy fact that every scattered adequate compact is Eberlein, which, in particular, gives a positive answer to Godefroy's question in the realm of spaces, adequate. We also show that has a 1-norming M-basis, whenever is adequate.
Recall that, given a set S, a family is adequate [75] if:
- (i)
for each ;
- (ii)
if and , then ;
- (iii)
if
Acknowledgements
The authors wish to express their gratitude to Marián Fabian and Gilles Godefroy for their insightful remarks on the problem considered in the article. Such remarks were extremely useful when preparing the final version of the manuscript.
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P. Hájek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. Research of T. Russo was supported by the GAČR project 20-22230L; RVO: 67985840 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. J. Somaglia was supported by Università degli Studi di Milano, Research Support Plan 2019 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. Research of S. Todorčević is partially supported by grants from NSERC (455916) and CNRS (UMR7586).