The Orthogonal Projection and the Riesz Representation Theorem

• Autorzy: Keiko Narita , Noboru Endou , Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 46E20: Hilbert spaces of continuous, differentiable or analytic functions
• 46C15: Characterizations of Hilbert spaces

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2015
• Tom: 23
• Numer: 3
• Strony: 243-252

Daty

wydano

2015-09-01

otrzymano

2015-07-01

online

2015-09-30

Twórcy

autor

Keiko Narita

• Hirosaki-city Aomori, Japan

autor

Noboru Endou

• Gifu National College of Technology Gifu, Japan

autor

Yasunari Shidama

• Shinshu University Nagano, Japan

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Identyfikatory

DOI

10.1515/forma-2015-0020