  We provide necessary and sufficient conditions for a $1$-jet $(f,
G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ for
some $F\in C^{1, \omega}(X)$. Here $E$ stands for an arbitrary subset of a
Hilbert space $X$ and $\omega$ is a modulus of continuity. As a corollary, in
the particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear)
operator whose norm does not depend on the dimension $n$. Furthermore, we
construct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear)
operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a natural
seminorm arising from the constants in the given condition for extension (and
the bounds we obtain are almost sharp); (2) $F$ is given by an explicit
formula; (3) $(F, \nabla F)$ depend continuously on the given data $(f, G)$;
(4) if $f$ is bounded (resp. if $G$ is bounded) then so is $F$ (resp. $F$ is
Lipschitz). We also provide similar results on superreflexive Banach spaces.

Functional Analysis;Classical Analysis and ODEs