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Erschienen in:
2018 | OriginalPaper | Buchkapitel
5. Linear Functionals
verfasst von : Sergei Ovchinnikov
Erschienen in: Functional Analysis
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Abstract
The concept of a linear functional is one of the main concepts in linear algebra and one that plays a significant role in functional analysis. In this chapter, we present basic facts of the theory of linear functionals on topological vector spaces. First, in Section 5.1, we describe bounded the linear functionals on sequence spaces introduced in Chapter 3. These results show that the dual spaces of these normed spaces are very rich. The natural question about the mere existence of nontrivial linear functionals is addressed by various Hahn–Banach theorems, which are found in Sections 5.2 and 5.3. An application of Hahn–Banach theorems is presented in Section 5.4, where all bounded linear functionals on the space C[a, b] are described. Finally, reflexive spaces and their elementary properties are introduced in Section 5.5.