Isomorphisms from the Space of Multilinear Operators

In this article, using the Mizar system [5], [2], the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators.The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to [6], [11], [9], [3], [10] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485–492, 1996.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Nelson Dunford and Jacob T. Schwartz. Linear operators I. Interscience Publ., 1958.Yuichi Futa, Noboru Endou, and Yasunari Shidama. Isometric differentiable functions on real normed space. Formalized Mathematics, 21(4):249–260, 2013. doi:10.2478/forma-2013-0027.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho. Bilinear operators on normed linear spaces. Formalized Mathematics, 27(1):15–23, 2019. doi:10.2478/forma-2019-0002.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Kosaku Yoshida. Functional Analysis. Springer, 1980.27210110

Invertible Operators on Banach Spaces

In this article, using the Mizar system [2], [1], we discuss invertible operators on Banach spaces. In the first chapter, we formalized the theorem that denotes any operators that are close enough to an invertible operator are also invertible by using the property of Neumann series.In the second chapter, we formalized the continuity of an isomorphism that maps an invertible operator on Banach spaces to its inverse. These results are used in the proof of the implicit function theorem. We referred to [3], [10], [6], [7] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Yasunari Shidama. The Banach algebra of bounded linear operators. Formalized Mathematics, 12(2):103–108, 2004.Kosaku Yoshida. Functional Analysis. Springer, 1980.27210711

Transformation Tools for Real Linear Spaces

This paper, using the Mizar system [1], [2], provides useful tools for working with real linear spaces and real normed spaces. These include the identification of a real number set with a one-dimensional real normed space, the relationships between real linear spaces and real Euclidean spaces, the transformation from a real linear space to a real vector space, and the properties of basis and dimensions of real linear spaces. We referred to [6], [10], [8], [9] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. Formalized Mathematics, 13(4):577–580, 2005.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81–85, 2007. doi:10.2478/v10037-007-0010-y.Takao Inoué, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation, differentiation and continuity on n-dimensional real normed linear spaces. Formalized Mathematics, 19(2):65–68, 2011. doi:10.2478/v10037-011-0011-8.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Kosaku Yosida. Functional Analysis. Springer, 1980.302939

Implicit Function Theorem. Part II

In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, JapanYasunari Shidama - Shinshu University, Nagano, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Hiroshi Imura, Morishige Kimura, and Yasunari Shidama. The differentiable functions on normed linear spaces. Formalized Mathematics, 12(3):321–327, 2004.Kazuhisa Nakasho. Invertible operators on Banach spaces. Formalized Mathematics, 27 (2):107–115, 2019. doi:10.2478/forma-2019-0012.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269–275, 2004.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Hideki Sakurai, Hiroyuki Okazaki, and Yasunari Shidama. Banach’s continuous inverse theorem and closed graph theorem. Formalized Mathematics, 20(4):271–274, 2012. doi:10.2478/v10037-012-0032-y.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.27211713

Bilinear Operators on Normed Linear Spaces

The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar [1]. In the first two chapters, algebraic structures [3] of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space.In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to [4], [11], [2], [7] and [8] in this formalization.Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Nelson Dunford and Jacob T. Schwartz. Linear operators I. Interscience Publ., 1958.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363–371, 2016. doi:10.15439/2016F520.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Continuity of bounded linear operators on normed linear spaces. Formalized Mathematics, 26(3):231–237, 2018. doi:10.2478/forma-2018-0021.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39–48, 2004.Yasumasa Suzuki, Noboru Endou, and Yasunari Shidama. Banach space of absolute summable real sequences. Formalized Mathematics, 11(4):377–380, 2003.Kosaku Yoshida. Functional Analysis. Springer, 1980.271152

Multilinear Operator and Its Basic Properties

In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure [2] of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to [3], [7], [5], [6] in this formalization.Yamaguchi University, Yamaguchi, JapanCzesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. On algebraic hierarchies in mathematical repository of Mizar. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, Proceedings of the 2016 Federated Conference on Computer Science and Information Systems (FedCSIS), volume 8 of Annals of Computer Science and Information Systems, pages 363–371, 2016. doi:10.15439/2016F520.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Marco Riccardi. Pocklington’s theorem and Bertrand’s postulate. Formalized Mathematics, 14(2):47–52, 2006. doi:10.2478/v10037-006-0007-y.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Kosaku Yoshida. Functional Analysis. Springer, 1980.271354

Riemann-Stieltjes Integral

In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties.In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].Narita Keiko - Hirosaki-city Aomori, JapanNakasho Kazuhisa - Akita Prefectural University Akita, JapanShidama Yasunari - Shinshu University Nagano, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1.Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.S.L. Gupta and Nisha Rani. Fundamental Real Analysis. Vikas Pub., 1986.Einar Hille. Methods in classical and functional analysis. Addison-Wesley Publishing Co., Halsted Press, 1974.H. Kestelman. Modern theories of integration. Dover Publications, 2nd edition, 1960.Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477-481, 1990.Keiichi Miyajima, Takahiro Kato, and Yasunari Shidama. Riemann integral of functions from ℝ into real normed space. Formalized Mathematics, 19(1):17-22, 2011.Daniel W. Stroock. A Concise Introduction to the Theory of Integration. Springer Science & Business Media, 1999

On Implicit and Inverse Function Theorems on Euclidean Spaces

Previous Mizar articles [7, 6, 5] formalized the implicit andinverse function theorems for Frechet continuously differentiable maps on Banachspaces. In this paper, using the Mizar system [1], [2], we formalize these theoremson Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in this formalization.Kazuhisa Nakasho - Yamaguchi University, Yamaguchi, JapanYasunari Shidama - Karuizawa Hotch 244-1, Nagano, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art andbeyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Vol-ker Sorge, editors, Intelligent Computer Mathematics, volume 9150 ofLecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615_817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32,2018. doi:10.1007/s10817-017-9440-6.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rⁿ. Formalized Mathematics, 15(2):65–72, 2007. doi:10.2478/v10037-007-0008-5.Miyadera Isao. Functional Analysis. Riko-Gaku-Sya, 1972.Kazuhisa Nakasho and Yuichi Futa. Inverse function theorem. Part I. Formalized Mathematics, 29(1):9–19, 2021. doi:10.2478/forma-2021-0002Kazuhisa Nakasho and Yasunari Shidama. Implicit function theorem. Part II. Formalized Mathematics, 27(2):117–131, 2019. doi:10.2478/forma-2019-0013.Kazuhisa Nakasho, Yuichi Futa, and Yasunari Shidama. Implicit function theorem. Part I. Formalized Mathematics, 25(4):269–281, 2017. doi:10.1515/forma-2017-0026.Kazuhisa Nakasho, Hiroyuki Okazaki, and Yasunari Shidama. Real vector space andrelated notions. Formalized Mathematics, 29(3):117–127, 2021. doi:10.2478/forma-2021-0012.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of familyof real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Laurent Schwartz. Calcul différentiel, tome 2. Analyse. Hermann, 1997.Kôsaku Yosida.Functional Analysis. Springer, 1980.30315916

The 3-Fold Product Space of Real Normed Spaces and its Properties

In this article, we formalize in Mizar [1], [2] the 3-fold product space of real normed spaces for usefulness in application fields such as engineering, although the formalization of the 2-fold product space of real normed spaces has been stored in the Mizar Mathematical Library [3]. First, we prove some theorems about the 3-variable function and 3-fold Cartesian product for preparation. Then we formalize the definition of 3-fold product space of real linear spaces. Finally, we formulate the definition of 3-fold product space of real normed spaces. We referred to [7] and [6] in the formalization.Hiroyuki Okazaki - Shinshu University, Nagano, JapanKazuhisa Nakasho, Yamaguchi University, Yamaguchi, JapanGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. The product space of real normed spaces and its properties. Formalized Mathematics, 15(3):81–85, 2007. doi:10.2478/v10037-007-0010-y.Artur Korniłowicz. Compactness of the bounded closed subsets of Ε²т. Formalized Mathematics, 8(1):61–68, 1999.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51–59, 2011. doi:10.2478/v10037-011-0009-2.Michael Read and Barry Simon. Functional Analysis (Methods of Modern Mathematical Physics). Academic Press, 1980.Kosaku Yosida. Functional Analysis. Springer, 1980.29424124