Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars
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Advances in Applied Clifford Algebras
Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars Heera Saini, Aditi Sharma and Romesh Kumar∗ Communicated by Irene Sabadini Abstract. We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved. Mathematics Subject Classification. 30G35, 46A22, 46A30. Keywords. Bicomplex modules, Hyperbolic modules, Topological bicomplex modules, Topological hyperbolic modules, Hyperbolic convexity, Hyperbolic-valued Minkowski functionals, Continuous linear functionals, Baire category theorem, Uniform boundedness principle, Open mapping theorem, Inverse mapping theorem, Closed graph theorem, Hahn Banach separation theorem.
1. Introduction and Preliminaries In this paper, we prove several core results of bicomplex functional analysis. The four basic principles, the Hahn Banach separation theorem for topological hyperbolic and topological bicomplex modules, the open mapping theorem, the closed graph theorem and the uniform boundedness principle for F -bicomplex and F -hyperbolic modules with hyperbolic-valued norm have been presented. Hahn Banach theorem is one of the most fundamental results in functional analysis. The theorem for real vector spaces was first proved by Hahn [13], rediscovered in its present, more general form by Banach [2]. Further, it was generalized to complex vector spaces by Bohnenblust and Sobczyk [4]; and Soukhomlinoff [32] who also considered the case of vector ∗ Corresponding
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Adv. Appl. Clifford Algebras
spaces with quaternionic scalars. The Hahn–Banach theorem for bicomplex functional analysis with real-valued norm was proved in [14]. Recently, Hahn Banach theorem for bicomplex functional analysis with hyperbolic-valued norm was proved in [19], which is in analytic form, involving the existence of extensions of a linear functionals. The uniform boundedness principle was proved by Banach and Steinhaus [3]. Some related results to this principle were already proved by Lebesgue [18], Hahn [12], Steinhaus [33], and Saks and Tamarkin [30]. The open mapping theorem was first proved in 1929 by Banach [2]. Later, in 1930, a different proof was given by Schauder [31]. There is a vast literature on generalizations of Hahn Banach theorem, the open mapping theorem and the uniform boundedness principle in different directions. The main idea of the paper is to extend the work of [19] to the more general case of topological hyperbolic and topological bicomplex modules. Both [14,19] have presented the Hahn Banach theorem in analtyic form with real-valued and hyperbolic-valued norm respectively. We are interested to present this result in its geometric form which has been done in Sects. 6,7 and 8. The work in this paper is organized in the following manner. In Sect
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