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A Generalization of the Riesz Representation Theorem to Infinite Dimensions

✍ Scribed by Yuh-Jia Lee

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
393 KB
Volume
151
Category
Article
ISSN
0022-1236

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✦ Synopsis

Let C p be the collection of real-valued functions f defined on E &p such that f is uniformly continuous on bounded subsets of

Then C is a complete countably normed space equipped with the family [&}& , p : p=1, 2, 3, ...] of norms. In this paper it is shown that to every bounded linear functional F in C* , there corresponds a signed measure & F such that F(.)= E* .(x) & F (dx) for . # C . It is also shown that there exists some p such that the measurable support of & is contained in E &p and E&p exp( 1 2 |x| 2 &p ) |& F | (dx)< . This result extends the Riesz representation theorem to infinite dimensions. In the course of the proof, an infinite dimensional analogue of the Weierstrass approximation theorem is also established on E*.

1997 Academic Press [3, 5 7] and the references cited there). In the present paper we shall generalize the Riesz representation theorem to infinite dimensions. The underlying infinite dimensional space under consideration will be taken to be the dual space E* of a nuclear space E. An important step towards such article no. FU973143

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