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Kadec norms on spaces of continuous functions. (English) Zbl 1164.46303

Summary: We study the existence of pointwise Kadec renormings for Banach spaces of the form \(C(K)\). We show in particular that such a renorming exists when \(K\) is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if \(C(K_1)\) has a pointwise Kadec renorming and \(K_2\) belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then \(C(K_1\times K_2)\) has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
46B26 Nonseparable Banach spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
54C35 Function spaces in general topology
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