Bilinear Operators on Normed Linear Spaces

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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.

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  • [5] 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.

  • [6] 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.

  • [7] Laurent Schwartz. Théorie des ensembles et topologie tome 1. Analyse. Hermann 1997.

  • [8] Laurent Schwartz. Calcul différentiel tome 2. Analyse. Hermann 1997.

  • [9] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics 12(1):39–48 2004.

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

  • [11] Kosaku Yoshida. Functional Analysis. Springer 1980.

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