Smooth surjections without surjective restrictions (Q375739)

Let \(E,F\) be real or complex Banach spaces and \(f:E\rightarrow F\) be a surjective map with some strong differentiability properties. The main goal of the paper is to investigate when there is a smaller Banach space \(G\) inside \(E\) such that the restriction of \(f\) to \(G\) remains surjective. Among other interesting results, the authors prove: (1) Let \(E\) be any infinite-dimensional Banach space and \(n\) a positive integer. Then there exists a \(C^{\infty}\)-smooth map \(f:E\rightarrow \mathbb{R}^{n}\) such that \(f\) is surjective and, when restricted to any finite dimensional subspace of \(E\), \(f\) is not surjective. (2) Let \(E\) be a Banach space with \(C^{m}\)-cellularity \(\geq2^{\aleph_{0}}\) and let \(F\) be a separable Banach space with dimension greater than \(1\). Then there exists a \(C^{m}\) smooth map \(f:E\rightarrow F\) such that \(f\) is surjective and, when restricted to any separable subspace of \(E\), \(f\) is not surjective. Several consequences of (2) are presented. The case of finite-dimensional Banach spaces is also considered.