The Stability of Input-Output Dynamical Systems (Enhanced Edition‪)‬

This chapter provides an introduction to the fundamental topological and
functional analytic concepts used extensively in this book. Input-output
stability is essentially a topological concept defined on spaces of time
functions which have various properties including those of linear topological
spaces. Thus in Section 1.2 the concept of linear spaces and various
topological notions associated with linear spaces are introduced. In Section
1.3 we develop generalized topological vector spaces with particular
structural forms of metric spaces, inductive and projective limits, and
Banach and Hilbert spaces. Various results in Chapters 5 and 6 depend
upon the fixed-point theorems introduced in Section 1.4. The majority of
time functions of interest in this book are L,, spaces and are reviewed as
part of Section 1.5 on measures and function spaces. Finally in Section
1.6 the concepts of functionals and dual spaces are developed.