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

Chapter 2Vector and function spaces

In Chapter, we loosely defined functional data to be a collection of sample paths for a stochastic process (or processes) with the index set . Thus, the data are functions that may have various properties such as being continuous or square integrable with probability one. Characteristics such as these signify a commonality that becomes amenable to treatment through the study of function spaces. An understanding of function spaces is an essential first step in dealing with functional data. Beyond that, the properties of linear functionals and operators on function spaces that are studied in Chapters 3 and 4 lie at the heart of the functional analogs of the basic concepts from multivariate analysis.

The purpose of this chapter is to present the function space theory that we perceive to be most relevant for fda. Clearly, it will be impossible to give a comprehensive treatment of each topic that we touch upon here. Certain function space concepts like reproducing kernel Hilbert spaces and Sobolev spaces are treated in some detail in view of their relative obscurity and relevance for our targeted audience. However, many topics such as vector, metric, Banach, and Hilbert spaces are included for completeness and to serve as a review of material that the reader will have hopefully seen elsewhere. Thorough expositions of these concepts can be found in, ...