Smooth functions on C(K)

Natural examples of spaces satisfying (i) or (ii) are c0 and the original Tsirelson space, and Bates asked whether indeed c0 / ∈ B. The question was settled in [9] (i.e. c0 / ∈ B), a paper which was conducted without any knowledge of S. Bates’ work, and which was mainly concerned with the behavior of C-smooth real functions on c0. In order to reveal the connection between these matters, let us denote by C the class of Banach spaces X such that for any real function f defined on an open subset U of X, with locally uniformly continuous derivative, f ′ is locally compact. That is to say, for every x ∈ U there exists open neighbourhood V ⊂ U , x ∈ V, such that f (V) is relatively compact in X. A simple use of the Baire category principle implies that if T : X → Y is a surjective operator with locally continuous derivative (e.g. C-Fréchet smooth), and X ∈ C, then Y ∈ C. If, on the other hand, Y ∈ B then X ∈ B. Since l2 ∈ B, l2 / ∈ C we have the following implication: X ∈ C =⇒ X / ∈ B, and moreover whenever Y ∈ B, there exists no surjective Y : X → Y with locally uniformly continuous derivative. A little more can be said under some additional assumptions.