Normed and Function Spaces

Houshang H. Sohrab
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Houshang H. Sohrab: Towson University, Mathematics Department

Chapter 9 in Basic Real Analysis, 2003, pp 351-393 from Springer

Abstract: Abstract Metric spaces were defined and studied in Chapter 5. Despite their importance, general metric spaces do not share the algebraic properties of the fields ℝ and ℂ, because a metric space need not be an algebra or even a vector space. Any (nonempty) subset of a metric space is again a metric space with the metric it borrows from the ambient space. Thus, curves and surfaces in the Euclidean space ℝ3 are metric spaces but are almost never vector subspaces of ℝ3. Even straight lines and planes are not vector subspaces unless they pass through the origin. There is an important class of metric spaces, however, that is a natural framework for the extension of topological as well as algebraic properties of ℝ and ℂ : It is the class of Normed Spaces, which we now define. Throughout this chapter, $$ \mathbb{F} $$ will stand for either ℝ or ℂ and X, Y, Z, etc. will denote vector spaces over $$ \mathbb{F} $$ .

Keywords: Hilbert Space; Banach Space; Normed Space; Banach Algebra; Cauchy Sequence (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-0-8176-8232-3_9

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