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
- Linear Space
- Normed Linear Space
- Linear Topological Space
- Continuous Linear Functional
- Real Linear Space
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References for Chapter IV
For the Hahn-Banach theorems and related topics, see Banach, S. [1] Théorie des Opérations Linéaires, Warszawa 1932.
Bourbaki, N. [2]
and Köthe, G. [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 The proof of Helly’s theorem given in this book is due to Y. Mimura (unpublished).
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© 1965 Springer-Verlag Berlin Heidelberg
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Yosida, K. (1965). The Hahn-Banach Theorems. In: Functional Analysis. Die Grundlehren der Mathematischen Wissenschaften, vol 123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-25762-3_5
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DOI: https://doi.org/10.1007/978-3-662-25762-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-23675-8
Online ISBN: 978-3-662-25762-3
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