Über Homöomorphe und Differenzierbare Abbildungen N. A. Banachräume

Let E and E′ be finite-dimensional n. a. Banach spaces over a complete n. a. non-trivial valued field K.

1. Let K be locally compact, dim(E)≤dim(E′) and f be a mapping from an open set of E into E'. Then we obtain an analogue of the classical theorem of Sard: The set of “critical values” of f is of outer Haar-measure zero.
2. If one of the following two conditions (i) K is locally compact and dim(E)>dim(E′), (ii) K is not locally compact, is satisfied, then there exists a differentiable homeomorphism f: E→E′ with a derivative vanishing everywhere on E.