  Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We
characterize compactness of $T$ in terms of differentiability of the Lipschitz
functions defined on $X$ with values in another normed space $Z$. Furthermore,
using a similar technique we can also characterize finite rank operators in
terms of differentiability of a wider class of functions but still with
Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On
the other hand, we give an extension of a result of Bourgain and Diestel
related to limited operators and cosingularity.
