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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

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

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Volume 23, Issue 3 (Sep 2015)


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


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


  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [3] Haim Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.Google Scholar

  • [4] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.Google Scholar

  • [5] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [6] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [7] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [9] Peter D. Dax. Functional Analysis. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley Interscience, 2002.Google Scholar

  • [10] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Subspaces and cosets of subspace of real unitary space. Formalized Mathematics, 11(1):1-7, 2003.Google Scholar

  • [11] Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Topology of real unitary space. Formalized Mathematics, 11(1):33-38, 2003.Google Scholar

  • [12] Noboru Endou, Yasumasa Suzuki, and Yasunari Shidama. Real linear space of real sequences. Formalized Mathematics, 11(3):249-253, 2003.Google Scholar

  • [13] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Google Scholar

  • [14] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Google Scholar

  • [15] Keiko Narita, Noboru Endou, and Yasunari Shidama. Dual spaces and Hahn-Banach theorem. Formalized Mathematics, 22(1):69-77, 2014. doi:10.2478/forma-2014-0007.CrossrefGoogle Scholar

  • [16] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Google Scholar

  • [17] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004.Google Scholar

  • [18] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29-34, 1993.Google Scholar

  • [19] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.Google Scholar

  • [20] Jan Popiołek. Introduction to Banach and Hilbert spaces - part I. Formalized Mathematics, 2(4):511-516, 1991.Google Scholar

  • [21] Jan Popiołek. Introduction to Banach and Hilbert spaces - part II. Formalized Mathematics, 2(4):517-521, 1991.Google Scholar

  • [22] Jan Popiołek. Introduction to Banach and Hilbert spaces - part III. Formalized Mathematics, 2(4):523-526, 1991.Google Scholar

  • [23] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991.Google Scholar

  • [24] Walter Rudin. Functional Analysis. New York, McGraw-Hill, 2nd edition, 1991.Google Scholar

  • [25] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004.Google Scholar

  • [26] Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377-380, 2003.Google Scholar

  • [27] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Google Scholar

  • [28] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Google Scholar

  • [29] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.Google Scholar

  • [30] Wojciech A. Trybulec. Subspaces and cosets of subspaces in real linear space. Formalized Mathematics, 1(2):297-301, 1990.Google Scholar

  • [31] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Google Scholar

  • [32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Google Scholar

  • [34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

  • [35] Hiroshi Yamazaki, Yasumasa Suzuki, Takao Inou´e, and Yasunari Shidama. On some properties of real Hilbert space. Part I. Formalized Mathematics, 11(3):225-229, 2003.Google Scholar

  • [36] Kosaku Yoshida. Functional Analysis. Springer, 1980. Google Scholar

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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