On the range of the derivatives of a smooth function between Banach spaces

Abstract

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces $X$ and $Y$ to ensure the existence of a $C^p$ smooth (Fréchet smooth or a continuous Gâteaux smooth) function $f$ from $X$ onto $Y$ such that $f$ vanishes outside a bounded set and all the derivatives of $f$ are surjections. In particular we deduce the following results. For the Gâteaux case, when $X$ and $Y$ are separable and $X$ is infinite-dimensional, there exists a continuous Gâteaux smooth function $f$ from $X$ to $Y$, with bounded support, so that $f^{\prime}(X) = {\cal L}(X,Y)$. In the Fréchet case, we get that if a Banach space $X$ has a Fréchet smooth bump and ${\rm dens} X = {\rm dens} {\cal L}(X,Y)$, then there is a Fréchet smooth function $f: X \rightarrow Y$ with bounded support so that $f^{\prime}(X) = {\cal L}(X,Y)$. Moreover, we see that if $X$ has a $C^p$ smooth bump with bounded derivatives and ${\rm dens} X = {\rm dens} {\cal L}^m_s(X;Y)$ then there exists another $C^p$ smooth function $f: X \rightarrow Y$ so that $f^{(k)} (X) = {\cal L}^k_s(X;Y)$ for all $k = 0, 1,\ldots, m$. As an application, we show that every bounded starlike body on a separable Banach space $X$ with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space $X^*$. In the non-separable case, we prove that $X$ has such property if $X$ has smooth partitions of unity.