Metric Spaces, Normed Spaces, Inner Product Spaces

Abstract

The goal of this chapter is to introduce the abstract theory of the spaces that are important in functional analysis and to provide examples of such spaces. These will serve as our examples throughout the rest of the text, and the spaces introduced in the second section of this chapter will be studied in great detail. The abstract spaces—metric spaces, normed spaces, and inner product spaces—are all examples of what are more generally called “topological spaces.” These spaces have been given in order of increasing structure. That is, every inner product space is a normed space, and in turn, every normed space is a metric space. It is “easiest,” then, to be a metric space, but because of the added structure, it is “easiest” to work with inner product spaces.KeywordsLinear SpaceNormed SpaceSequence SpaceProduct SpaceNormed Linear SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.