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Smooth functions on \(c_0\). (English) Zbl 0940.46023
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 H. Torunczyk, Stud. Math. 46, 43-51 (1973; Zbl 0251.46022) and 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.
Reviewer: A.Kriegl (Wien)

46G05 Derivatives of functions in infinite-dimensional spaces
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