  Projection operators are important in Analysis, Optimization and Algorithm.
It is well known that these operators are firmly nonexpansive. In this paper,
we provide an exact result that sharpens this well-known result. We develop the
theory of averaged operators and provide a lower bound. We give a result on the
avergedness of operator compositions. We also provide some nonlinear examples
to illustrate our results.

Functional Analysis;Numerical Analysis;Numerical Analysis;Optimization and Control