This book is a self-contained account of knowledge of the theory of
nonlinear superposition operators: a generalization of the notion
of functions. The theory developed here is applicable to operators
in a wide variety of function spaces, and it is here that the
modern theory diverges from classical nonlinear analysis. The
purpose of this book is to collect the basic facts about the
superposition operator, to present the main ideas which are useful
in studying its properties and to provide a comparison of its
behaviour in different function spaces. Some applications are also
considered, for example to control theory and optimization. Much of
the work here has only appeared before in research literature,
which itself is catalogued in detail here.
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