Abstract
In Section 1 we introduce normed linear spaces and bounded linear mappings. Section 2 is concerned with finite-dimensional spaces and with the problem of best approximation by elements of a finite-dimensional subspace. In Section 3 we discuss Lp spaces; we prove the completeness of Lp, and determine the form of the normable linear functionals on Lp in case p> 1. (In contrast to the classical theory, a bounded linear functional need not have a norm.) We then apply these results to the proof of the Radon-Nikodym theorem.
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© 1985 Springer-Verlag Berlin Heidelberg
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Bishop, E., Bridges, D. (1985). Normed Linear Spaces. In: Constructive Analysis. Grundlehren der mathematischen Wissenschaften, vol 279. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61667-9_8
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DOI: https://doi.org/10.1007/978-3-642-61667-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64905-9
Online ISBN: 978-3-642-61667-9
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