For many decades great mathematical interest has focused on problems
associated with linear operators and the extension of the well-known results
of linear algebra to an infinite-dimensional context. This interest has
been crowned with deep insights, and the substantial theory that has been
developed has had a profound influence throughout the mathematical sciences.
However when one drops the assumption of linearity, the associated
operator theory and the many concrete problems associated with
such a theory represent a frontier of mathematical research. Nonetheless,
the fundamental results so far obtained in this direction already form a
deep and beautiful extension of this linear theory. Just as in the linear
case, these results were inspired by and are highly relevant to concrete
problems in mathematical analysis. The object of the lectures represented
here is a systematic description of these fundamental nonlinear results and
their applicability to a variety of concrete problems taken from various
fields of mathematical analysis.