  Motivated by the Sard conjecture in sub-Riemannian geometry, and more
specifically on Carnot groups, we consider the set of maps defined on a Hilbert
space $H$: \[\mathscr{P}_{d}^m(H)=\bigg\{f:H\to \mathbb{R}^m\,\bigg|\,
\forall\, E\subset H, \,\dim(E)<\infty \implies f|_{E} \textrm{ is a polynomial
map of degree $d$}\bigg\}.
  \] Example of maps belonging to this class are: the map $f:\ell^2\to
\mathbb{R}$ constructed by Kukpa as a counterexample for the Sard theorem in
infinite dimension; the Endpoint maps of Carnot groups.
  We prove that, for infinite-dimensional subspaces $V \subset H$ satisfying
quantitative assumptions on their Kolmogorov $n$-width, the restriction of $f$
on $V$ has the Sard property, i.e. the set of critical values of $f|_{V}$ has
measure zero in $\mathbb{R}^m$. We prove various quantitative versions of this
result, allowing to study the Sard property also for the case $V=H$. By
constructing suitable examples, we prove the sharpness of our $n$-width
assumptions.
  We apply these results to sub-Riemannian geometry, proving that on Carnot
groups the Sard property holds true for the restriction of the Endpoint map on
the set of real-analytic controls with large enough radius of convergence. This
is a more functional-analytic approach to the Sard conjecture that, unlike
previous ones, does not resort to reduction to finite-dimensions.
