The Construction of Hilbert Spaces over the Non-Newtonian Field

Abstract

Although there are many excellent ways to present the principle of the classical calculus, the novel presentations probably lead most naturally to the development of the non-Newtonian calculi. In this paper we introduce vector spaces over real and complex nonNewtonian field with respect to the * -calculus which is a branch of non-Newtonian calculus. Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complete inner product spaces in the sense of * -calculus. Furthermore, as an example of Hilbert spaces, first we introduce the non-Cartesian plane which is a nonlinear model for plane Euclidean geometry. Secondly, we give Euclidean, unitary, and sequence spaces via corresponding norms which are induced by an inner product. Finally, by using the * -norm properties of complex structures, we examine CauchySchwarz and triangle inequalities.

On multiplicative difference sequence spaces and related dual properties

“…Also some authors have also worked on the classical sequence spaces and related topics by using this type calculus [14,15]. Further Kadak et al [16,17,18] have examined matrix transformations between certain sequence spaces and have generalized Runge-Kutta numerical method. …”

Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus

“…Also some authors have also worked on the classical sequence spaces and related topics by using non-Newtonian calculus [16][17][18]. Further Kadak [19] and Kadak and Efe [20,21] have determined Kothe-Toeplitz duals and matrix transformations between certain sequence spaces over the non-Newtonian complex field.…”

On the Classical Paranormed Sequence Spaces and Related Duals over the Non-Newtonian Complex Field

“…Also some authors have also worked on classical sequence spaces and related topics by using non-Newtonian calculus [18,19]. Further Kadak [20] and Kadak et al [21][22][23] have determined Kothe-Toeplitz duals and matrix transformations between certain sequence spaces over the non-Newtonian complex field and have generalized Runge-Kutta method with respect to the non-Newtonian calculus.…”