Ranges of operators and derivatives (Q961074)

The authors develop a method of constructing differentiable and Lipschitz functions between Banach spaces. They show that their method provides a unified way to prove results of \textit{S.\,M.\thinspace Bates} [Isr.\ J.\ Math.\ 100, 209--220 (1997; Zbl 0898.46044)], \textit{D.\,Azagra} and \textit{R.\,Deville} [J.~Funct.\ Anal.\ 180, No.\,2, 328--346 (2001; Zbl 0983.46016)] and \textit{D.\,Azagra, M.\,Jiménez-Sevilla} and \textit{R.\,Deville} [Math.\ Proc.\ Camb.\ Philos.\ Soc.\ 134, No.\,1, 163--185 (2003; Zbl 1034.46039)]. They also obtain some new results under special set theoretic assumptions. Using results of Todorčević, they prove that, under Martin's Maximum axiom: {\parindent5mm \begin{itemize}\item[(1)] For every two Banach spaces \(X\) and \(Y\) of density \(\aleph_1\), there exists a \(C^1\) Fréchet smooth and Lipschitz mapping from \(X\) onto \(Y\). \item[(2)] For every Asplund Banach space \(X\) of density \(\aleph_1\), there exists a \(C^1\)-Fréchet smooth and Lipschitz function \(f:X\longrightarrow\mathbb{R}\) whose range of the derivative contains all \(B_{X^\ast}\). \end{itemize}} Under the weaker axiom MA\(_\aleph{}_1{}\): {\parindent5mm \begin{itemize}\item[(3)] There is no surjective \(C^1\)-smooth function with locally uniformly continuous derivative from \(c_0(\omega_1)\) onto \(\ell_2\). \end{itemize}} The authors notice that the last statement fails under CH. However, they do not know whether the results (1) and (2) above hold in ZFC. They even ask whether every two Banach spaces of density \(\mathfrak c\) are the \(C^\infty\)-smooth image of each other.