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On the existence of overcomplete sets in some classical nonseparable Banach spaces. (English) Zbl 1476.46025

A subset \(Y\) of a Banach space \(X\) is overcomplete if \(|Y|= \mathrm{dens} (X)\) and \(Z\) is linearly dense in \(X\) for every \(Z\subseteq Y\) with \(|Z|=|Y|\). This notion was recently introduced by T. Russo and J. Somaglia [Proc. Am. Math. Soc. 149, No. 2, 701–714 (2021; Zbl 1456.46020)] as a generalisation of the notion of overcomplete sequence that has been present in the literature since the fifties. In the present paper, the author studies the problem of the existence of overcomplete sets in non-separable Banach spaces and obtains several rather general existence, non-existence, and independence results.
Instead of describing all results of the paper here (they are clearly listed in the Introduction of the paper), I will point out some of the most remarkable ones. First, the author proves that every WLD Banach space of density \(\omega_1\), as well as \(C([0,\omega_1])\), admits an overcomplete set. Concerning non-existence results, the author proves that the following Banach spaces admit no overcomplete sets: \(C(K)\) where \(K\) is infinite and extremally disconnected, \(\ell_\infty(\lambda)\), \(\ell_\infty(\lambda)/c_0(\lambda)\), \(L_\infty \{0,1\}^\lambda\) for any infinite cardinal, \(C([0,1]^\kappa)\), \(C(\{0,1\}^\kappa)\) if \(\kappa\) has uncountable cofinality. All these results follow from Theorem 29: if \(X\) contains \(\ell_1(\mathrm{dens}(X))\) and \(\mathrm{cf}(\mathrm{dens}(X))> \omega\), then \(X\) contains no overcomplete set. Moreover, under the assumption of \((\mathsf{MA}+\neg\mathsf{CH})\), the author proves, for example, the following: if \(X\) is a Banach space with \(\mathrm{dens}(X)< \mathfrak{c}\), \(\mathrm{cf}(\mathrm{dens}(X))> \omega\) and such that \((B_{X^*},w^*)\) is separable, then \(X\) contains no overcomplete set (Theorem 20).
Apart from the many general results which are proven, the paper is also of interest for the methods of proofs and for the many open questions that are listed in the last section. Just to name some: Assume that Banach spaces \(X\) and \(Y\) admit overcomplete sets; does \(X\oplus Y\) admit overcomplete sets as well? (This is open even in the case \(Y=\mathbb{R}\).) Assume that a Banach space \(X\) admits overcomplete sets and \(Y\subseteq X\) has the same density as \(X\); does \(Y\) admit overcomplete sets? Can one prove (in \(\mathsf{ZFC}\)) that, if a Banach space \(X\) admits an overcomplete set, then its density is at most \(\omega_1\)?

MSC:

46B26 Nonseparable Banach spaces
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
00A27 Lists of open problems

Citations:

Zbl 1456.46020
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References:

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