Smooth functions on \(c_0\)
It is proved that if the derivative of a Fréchet differentiable real function on the Banach space \(c_0\) is locally uniformly continuous then it is locally compact (i.e. maps a neighborhood to a relatively compact subset of \(\ell^1\)). The intuitive meaning of this statement is, that every \(C^2\)-function on \(c_0\) depends locally only on ``almost'' finitely many coordinates. See \textit{H. Torunczyk}, Stud. Math. 46, 43-51 (1973; Zbl 0251.46022) and \textit{R. Haydon}, C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1175-1178 (1992; Zbl 0788.46008) for the importance these functions on \(c_0\) play for smoothness properties of Banach spaces. Among the corollaries given in this paper answers to questions posed by J. A.~Jaramillo and by S.~Bates are given.
scientific article; zbMATH DE number 1251441