Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 22, No 1, (1996), 25p-28p
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
It was Haydon who first showed in  that C(K) spaces for K countable admit
an equivalent C∞-Fréchet smooth norm. Later, in  and  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  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.