Overcomplete sets in non-separable Banach spaces. (English) Zbl 1456.46020
In [Math. Scand. 6, 189–199 (1958; Zbl 0088.08502)], V. Klee introduced the notion of {overcomplete sequence}, meaning a sequence every subsequence of which is linearly dense. It is well known that overcomplete sequences exist in any separable Banach space. A sequence is called {almost overcomplete} whenever the closed linear span of each of its subsequences has finite codimension. It was proved in [V. P. Fonf and C. Zanco, J. Math. Anal. Appl. 420, No. 1, 94–101 (2014; Zbl 1310.46015)] that every {almost overcomplete bounded sequence} has to be relatively norm-compact. A simpler proof of this result is given at the end of this paper.
The authors introduce the notion of overcomplete set in a natural way. Unlike the separable case, it is proved that, if \(\operatorname {dens} X \geq \omega_2\), then \(X\) contains no overcomplete set, as well as that \(\ell_1(\Gamma)\) does not contain any overcomplete set whenever \(\Gamma\) is uncountable. On the other hand, under (CH) it is proved that every Banach space \(X\) such that \(\operatorname{dens} X = \operatorname{dens} X^*=\omega_1\) contains an overcomplete set. Recently, \textit{P. Koszmider} proved that in ZFC all WLD Banach spaces of density \(\omega_1\) admit overcomplete sets [``On the existence of overcomplete sets in some classical nonseparable Banach spaces'', Preprint (2020), arXiv:2006.00806]. Certainly, overcomplete sets in non-separable Banach spaces fail to be relatively norm-compact (unlike the aforementioned Fonf-Zanco result). However, it is proved that, under (CH), every WCG Banach space with density \(\omega_1\) contains a relatively weakly compact overcomplete set. The paper is well written and a joy to read\)
The authors introduce the notion of overcomplete set in a natural way. Unlike the separable case, it is proved that, if \(\operatorname {dens} X \geq \omega_2\), then \(X\) contains no overcomplete set, as well as that \(\ell_1(\Gamma)\) does not contain any overcomplete set whenever \(\Gamma\) is uncountable. On the other hand, under (CH) it is proved that every Banach space \(X\) such that \(\operatorname{dens} X = \operatorname{dens} X^*=\omega_1\) contains an overcomplete set. Recently, \textit{P. Koszmider} proved that in ZFC all WLD Banach spaces of density \(\omega_1\) admit overcomplete sets [``On the existence of overcomplete sets in some classical nonseparable Banach spaces'', Preprint (2020), arXiv:2006.00806]. Certainly, overcomplete sets in non-separable Banach spaces fail to be relatively norm-compact (unlike the aforementioned Fonf-Zanco result). However, it is proved that, under (CH), every WCG Banach space with density \(\omega_1\) contains a relatively weakly compact overcomplete set. The paper is well written and a joy to read\)
Reviewer: Daniele Puglisi (Catania)
MSC:
| 46B26 | Nonseparable Banach spaces |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
| 46A35 | Summability and bases in topological vector spaces |
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