On the range of the derivative of a smooth mapping between Banach spaces

Abstract

We survey recent results on the structure of the range of the derivative of a smooth mapping $f$ between two Banach spaces $X$ and $Y$. We recall some necessary conditions and some sufficient conditions on a subset $A$ of $ℒ\left(X,Y\right)$ for the existence of a Fréchet differentiable mapping $f$ from $X$ into $Y$ so that $f^{\prime }\left(X\right)=A$. Whenever $f$ is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping $f$ from ${\ell }^1\left(\mathbb{N}\right)$ into ${ℝ}^2$, which is bounded, Lipschitz-continuous, and so that for all $x,y\in {\ell }^1\left(\mathbb{N}\right)$, if $x\ne y$, then $\Vert f^{\prime }\left(x\right)-f^{\prime }\left(y\right)\Vert >1$.