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

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