This is an announcement for the paper "On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces" by M. Jimenez-Sevilla and L. Sanchez-Gonzalez.

Abstract: Let $X$ and $Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A \to Z$. We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \to Z$ such that $F_{\mid_A}=f$, when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$, or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty$.

Archive classification: math.FA

Mathematics Subject Classification: 46B20

Remarks: 17 pages

Submitted from: lfsanche@mat.ucm.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1112.5888

or

http://arXiv.org/abs/1112.5888