  If $L$ is a bounded linear operator mapping the Banach space $X$ into the
Banach space $Y$ and $K$ is a compact set in $X$, then the Kolmogorov widths of
the image $L(K)$ do not exceed those of $K$ multiplied by the norm of $L$. We
extend this result from linear maps to holomorphic mappings $u$ from $X$ to $Y$
in the following sense: when the $n$ widths of $K$ are $O(n^{-r})$ for some
$r\textgreater{}1$, then those of $u(K)$ are $O(n^{-s})$ for any $s \textless{}
r-1$, We then use these results to prove various theorems about Kolmogorov
widths of manifolds consisting of solutions to certain parametrized PDEs.
Results of this type are important in the numerical analysis of reduced bases
and other reduced modeling methods, since the best possible performance of such
methods is governed by the rate of decay of the Kolmogorov widths of the
solution manifold.
