Chapter 13 - Distances in Functional Analysis

Abstract

This chapter focuses on concept of distances in functional analysis. Functional Analysis is the branch of mathematics, concerned with the study of spaces of functions. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. In the modern view, functional analysis is seen as the study of complete normed vector spaces, that is, Banach spaces. For any real number p > 1, an example of a Banach space is given by Lp-space of all Lebesgue-measurable functions whose absolute value's pth power has finite integral. A Hilbert space is a Banach space in which the norm arises from an inner product. Also, in functional analysis are considered the continuous linear operators defined on Banach and Hilbert spaces. The chapter discusses the concepts related to metrics on function spaces including integral metric, uniform metric, dogkeeper distance, and Bohr metric. Details of metrics on linear operators are also explained in the chapter.