Weak bump functions and applications (Q2777728)

\textit{S. M. Bates} [Isr. J. Math. 100, 209-220 (1997; Zbl 0898.46044)] showed that if a Banach space \(X\) satisfies a certain technical condition and \(Y\) is any separable Banach space, then there is a \(C^\infty\) map \(g\) from \(X\) onto \(Y\). This is generalized here by weakening the technical condition and choosing \(g\) so that its derivative has rank one everywhere. This further exemplifies the failure of Rolle's theorem and of Sard's theorem in infinite dimensions. The main technique is to assume that \(X\) admits a suitable auxiliary norm and then to construct a bump function with respect to that weaker norm.