In 1909, F. Riesz succeeded in giving an integral represntation for continuous linear functionals on C[0,1]. Although other authors, notably Hadamard and Frechet, had given representations for continuous linear functionals on C[0,1], their results lacked the clarity, elegance, and some of the substance (uniqueness) of Riesz's theorem. Subsequently, the integral representation of continuous linear functionals has been known as the Riesz Representation Theorem. In this paper, three different proofs of the Riesz Representation Theorem are presented. The first approach uses the denseness of the Bernstein polynomials in C[0,1] along with results of Helly to write the continuous linear functionals as Stieltjes integrals. The second approach makes use of the Hahn-Banach Theorem in order to write the functional as an integral. The paper concludes with a detailed presentation of a Daniell integral development of the Riesz Representation Theorem.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc504232 |
| Date | 08 1900 |
| Creators | Williams, Stanley C. (Stanley Carl) |
| Contributors | Lewis, Paul Weldon, Hagan, Melvin R. |
| Publisher | North Texas State University |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | 72 leaves, Text |
| Rights | Public, Williams, Stanley C. (Stanley Carl), Copyright, Copyright is held by the author, unless otherwise noted. All rights reserved. |
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