  The purpose of this paper is to study the approximation of vector valued
mappings defined on a subset of a normed space. We investigate Korovkin-type
conditions under which a given sequence of linear operators becomes a so-called
approximation process. First, we give a sufficient condition for this sequence
to approximate the class of bounded, uniformly continuous functions. Then we
present some sufficient and necessary conditions guaranteeing the approximation
within the class of unbounded, *weak-to-norm continuous mappings. We also
derive some estimates of the rate of convergence.
