Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Chapter 3Linear operator and functionals

In Chapter 2, we provided a review of the basic facts concerning Banach and Hilbert spaces. What is of more interest for our purposes is the properties of functions that operate on such space. This chapter proceeds in that direction.

We are primarily interested in transformations that are linear and these tend to come in two varieties: linear functionals and operators, with the former being a special case of the latter. Both topics fall into the realm of mathematics known as functional analysis. This is a very broad area and our exposition cannot hope to do it justice. Rather than attempt to do so, we pick and choose topics that we feel are most relevant to fda and, in particular, those that are needed for subsequent chapters.

More complete treatments of functional analysis are available through many sources. For example, Dunford and Schwarz (1988) is a standard reference for linear operator theory; for an elementary introduction, one can consult the text by Rynne and Youngson (2001).

Before proceeding, it is perhaps worthwhile to comment on our use of the word “functional” as a adjective modifier of both “data” and “analysis” throughout this text. Functional analysis derives its name from its foundation in the study of linear functionals on, typically, Banach spaces. Such spaces need have nothing to do with functions per se. In contrast, functional data is, by definition, a collection of (random) functions that need have no direct connection ...