Fundamentals of Matrices and Operators

In this chapter, we collect some basic facts about functional analysis that are used throughout the book. By an operator, we mean a bounded linear operator. All linear spaces in this book are assumed to be over the field of complex numbers, denoted as \(\mathbb {C}\) . It is well known that there are significant differences between vector space structures over real and complex fields; see [1]. To simplify the notation, we frequently use the two-point compactification \(\mathbb {R}\cup \{-\infty ,+\infty \}\) and denote \(\infty \) as \(+\infty \) when there is no ambiguity.