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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
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Volume 23, Issue 3 (Sep 2015)

Issues

The Orthogonal Projection and the Riesz Representation Theorem

Keiko Narita / Noboru Endou / Yasunari Shidama
Published Online: 2015-09-30 | DOI: https://doi.org/10.1515/forma-2015-0020

Abstract

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces.

Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization.

Keywords: normed linear spaces; Banach spaces; duality; orthogonal projection; Riesz representation

MSC: 46E20; 46C15; 03B35

MML identifier:: DUALSP04

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About the article

Received: 2015-07-01

Published Online: 2015-09-30

Published in Print: 2015-09-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2015-0020.

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© by Keiko Narita. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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