  This paper aims at characterizing the approximability of bounded sets in the
range of nonlinear operators in Banach spaces by finite-dimensional linear
varieties. In particular, the class of operators we consider includes the
endpoint maps of nonlinear distributed-parameter control systems. We describe
the relationship between the Kolmogorov n-width of a bounded subset and the
width of its image under an essentially nonlinear transformation. We propose
explicit estimates of the n-width in the space of images in terms of the affine
part of the corresponding operator and the width of its nonlinear perturbation.
These $n$-width estimates enable us to describe the reachable sets for
infinite-dimensional bilinear control systems, with applications to controlling
the Euler-Bernoulli beam using a contraction force and to a single-input
Schr\"odinger equation.
