Metric Spaces, Normed Spaces, Inner Product Spaces
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 Space Normed Space Sequence Space Product Space Normed Linear Space
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- 2.Named for Viktor Bunyakovskii (1804–1889; Ukraine), Augustin Louis Cauchy (1789–1857; France), and Hermann Schwarz (1843–1921; Poland, now Germany).Google Scholar
- 4.In the Introduction we stated that the phrase “functional analysis” was coined by Lévy. Some authors suggest that the true inspiration for this phrase comes from Fréchet’s thesis. However, Fréchet himself credits Lévy (see page 260 of ).Google Scholar
- 5.Erik Ivar Fedholm (1866–1927) was a Swedish mathematician. You will read more about his work in Chapter 5. The others you have already encounteredGoogle Scholar