On the separable quotient problem for Banach spaces. (English) Zbl 1426.46012
Summary: While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces \(c_{0}\) and \(\ell _{p}\), for \(1\leq p\leq \infty \). Also recent results of S. A. Argyros et al. [Math. Ann. 341, No. 1, 15–38 (2008; Zbl 1160.46005)] and W. Śliwa [J. Math. Soc. Japan 64, No. 2, 387–397 (2012; Zbl 1253.46025)] are provided. This makes our presentation supplementary to a previous survey [Rev. Mat. Univ. Complutense Madr. 10, No. 2, 299–330 (1997; Zbl 0908.46007)] due to J. Mujica.
MSC:
| 46B26 | Nonseparable Banach spaces |
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46E27 | Spaces of measures |
| 46E40 | Spaces of vector- and operator-valued functions |
Keywords:
Banach space; barrelled space; separable quotient; vector-valued function space; linear operator space; vector measure space; tensor product; Radon-Nikodým propertyReferences:
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