In this chapter, we gather some basic facts about complex normed linear spaces and their operators. In particular, we discuss Banach spaces, Hilbert spaces and their bounded operators. There is no doubt that the subject is very vast and it is impossible to give a comprehensive treatment in one chapter. Our goal is to recall a few important aspects of the theory that are used in the study of H(b) spaces. We start by giving some examples of Banach spaces and introduce some classic operators. Then the dual space is defined and the well-known Hahn–Banach theorem is stated without proof. However, some applications of this essential result are outlined. Then we discuss the open mapping theorem (Theorem 1.14), the inverse mapping theorem (Corollary 1.15), the closed graph theorem (Corollary 1.18) and the uniform boundedness principle (Theorem 1.19). The common root of each of these theorems stems from the Baire category theorem (Theorem 1.13). Then we discuss Banach algebras and introduce the important concept of spectrum and state a simple version of the spectral mapping theorem (Theorem 1.22). At the end, we focus on Hilbert spaces, and some of their essential properties are outlined. We talk about Parseval's identity, the generalized version of the polarization identity, and Bessel's inequality. We also discuss in detail the compression of an operator to a closed subspace. Then we consider several topologies that one may face on a Hilbert space or on the space of its operators. The important concepts of adjoint and tensor product are discussed next. The chapter ends with some elementary facts about invariant subspaces and the cyclic vectors.

Banach spaces

Throughout this text we will consider only complex normed linear spaces. A complete normed linear space is called a Banach space. The term linear manifold refers to subsets of a linear space that are closed under the algebraic operations, while the term subspace is reserved for linear manifolds that are also closed in the norm or metric topology. Nevertheless, the combination closed subspace can also be found in the text. Given a subset ε of a Banach space X, we denote by Lin (ε) the linear manifold spanned by ε, which is the linear space whose elements are finite linear combinations of elements of ε.