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Petr Hájek, Richard Haydon, SMOOTH NORMS AND APPROXIMATION IN BANACH SPACES OF THE TYPE 𝒞(K), The Quarterly Journal of Mathematics, Volume 58, Issue 2, June 2007, Pages 221–228, https://doi.org/10.1093/qmath/ham010
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Abstract
Two results are proved about the Banach space X = 𝒞(K), where K is compact and Hausdorff. The first concerns smooth approximation: let m be a positive integer or ∞; we show that if there exists on X a non-zero function of class 𝒞m with bounded support, then all continuous real-valued functions on X can be uniformly approximated by functions of class 𝒞m. The second result is that if X admits a norm, equivalent to the supremum norm, with locally uniformly convex dual norm, then X also admits an equivalent norm that is of class 𝒞∞ (except at 0).