Abstract
In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.
Keywords
- Normed Linear Space
- Linear Topological Space
- Continuous Linear Functional
- Real Linear Space
- Complex Linear Space
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References for Chapter IV
For the Hahn-Banach theorems and related topics, see Banach [1], Bourbaki [2] and Köthe [1]. It was Mazur [2] who noticed the importance of convex sets in normed linear spaces. The proof of Helly’s theorem given in this book is due to Y. Mimura (unpublished).
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© 1974 Springer-Verlag Berlin Heidelberg
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Yosida, K. (1974). The Hahn-Banach Theorems. In: Functional Analysis. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96208-0_5
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DOI: https://doi.org/10.1007/978-3-642-96208-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-96210-3
Online ISBN: 978-3-642-96208-0
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