Geodesic
Analytic Renormings of C(K) Spaces
Serdica Mathematical Journal, Tome 22 (1996) no. 1, pp. 25-28

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The aim of our present note is to show the strength of the existence of an equivalent analytic renorming of a Banach space, even compared to C∞-Fréchet smooth renormings. It was Haydon who first showed in [8] that C(K) spaces for K countable admit an equivalent C∞-Fréchet smooth norm. Later, in [7] and [9] he introduced a large clams of tree-like (uncountable) compacts K for which C(K) admits an equivalent C∞-Fréchet smooth norm. Recently, it was shown in [3] that C(K) spaces for K countable admit an equivalent analytic norm. Our Theorem 1 shows that in the class of C(K) spaces this result is the best possible.
Keywords: Analytic Renormings 
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Hájek, Petr. Analytic Renormings of C(K) Spaces. Serdica Mathematical Journal, Tome 22 (1996) no. 1, pp. 25-28. http://geodesic.mathdoc.fr/item/SMJ2_1996_22_1_a1/