  The question which led to the title of this note is the following:
  {\it If $X$ is a Banach space and $K$ is a compact subset of $X$, is it
possible to find a compact, or even approximable, operator $v:X\to X$ such that
$K\subset\ol{v(B_X)}$?}
  This question was first posed by P.G.Dixon [6] in connection with
investigating the problem of the existence of approximate identities in certain
operator algebras. We shall provide a couple of observations related to the
above question and give in particular a negative answer in case of approximable
operators.
  We shall also provide the first examples of Banach spaces having the
approximation property but failing the bounded compact approximation property
though all of their duals do even have the metric compact approximation
property.
