A note on Banach spaces \(E\) for which \(E_w\) is homeomorphic to \(C_p(X)\). (English) Zbl 1507.46004
Summary: \(C_p(X)\) denotes the space of continuous real-valued functions on a Tychonoff space \(X\) endowed with the topology of pointwise convergence. A Banach space \(E\) equipped with the weak topology is denoted by \(E_w\). It is unknown whether \(C_p(K)\) and \(C(L)_w\) can be homeomorphic for infinite compact spaces \(K\) and \(L\) [M. Krupski, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 110, No. 2, 557–563 (2016; Zbl 1362.54017);
M. Krupski and W. Marciszewski, J. Math. Anal. Appl. 452, No. 1, 646–658 (2017; Zbl 1376.54019)]. In this paper we deal with a more general question: does there exist a Banach space \(E\) such that \(E_w\) is homeomorphic to the space \(C_p(X)\) for some infinite Tychonoff space \(X\)? We show that if such homeomorphism exists, then (a) \(X\) is a countable union of compact sets \(X_n\), \(n\in\omega\), where at least one component \(X_n\) is non-scattered; (b) the Banach space \(E\) necessarily contains an isomorphic copy of the Banach space \(\ell_1\).
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 54C35 | Function spaces in general topology |
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