  Metric projection operators can be defined in similar wayin Hilbert and
Banach spaces. At the same time, they differ signifitiantly in their
properties. Metric projection operator in Hilbert space is a monotone and
nonexpansive operator. It provides an absolutely best approximation for
arbitrary elements from Hilbert space by the elements of convex closed sets .
This leads to a variety of applications of this operator for investigating
theoretical questions in analysis and for approximation methods. Metric
projection operators in Banach space do not have properties mentioned above and
their applications are not straightforward. Two of the most important
applications of the method of metric projection operators are as follows:
  1. Solve a variational inequality by the iterative-projection method,
  2. Find common point of convex sets by the iterative-projection method.
  In Banach space these problems can not be solved in the framework of metric
projection operators. Therefore, in the present paper we introduce new
generalized projection operators in Banach space as a natural generalization of
metric projection operators in Hilbert space. In Sections 2 and 3 we introduce
notations and recall some results from the theory of variational inequalities
and theory of approximation. Then in Sections 4 and 5 we describe the
properties of metric projection operators $P_\Omega$ in Hilbert and Banach
spaces and also formulate equivalence theorems between variational inequalities
and direct projection equations with these
