The Construction of Hilbert Spaces over the Non-Newtonian Field

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 non-Newtonian 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 Cauchy-Schwarz and triangle inequalities. 1. Introduction The foundation of the theory of Hilbert spaces was laid down in 1912, D. Hilbert (1862–1943), on integral equations. However, an axiomatic basis of the theory was provided by J. Von Neumann (1903–1957). Since then this topic has become one of the most interesting and powerful subjects. Moreover, Hilbert spaces are the simplest type of infinite dimensional Banach spaces to play a remarkable role in functional analysis. Non-Newtonian calculus is an alternative to the usual calculus of Newton and Leibniz. It provides differentiation and integration tools based on non-Newtonian operations instead of classical operations. Every property in classical calculus has an analogue in non-Newtonian calculus. Generally, non-Newtonian calculus is a methodology that allows one to have a different look at problems which can be investigated via calculus. Furthermore the bigeometric calculus, which was created by Katz and Grossman in August 1970 as a branch of non-Newtonian calculus, has a derivative that is scale-free; that is, it is invariant under all changes of scales or units in function arguments and values. In some cases, for example, for wage-rate (in dollars, euro, etc.) related problems, the use of bigeometric calculus is advocated instead of a traditional Newtonian one (cf. [1–3]). Bashirov et al. [4, 5] have recently concentrated on the multiplicative calculus and gave the results with applications corresponding to the well-known properties of derivatives and integrals in the classical calculus. Uzer [6] has extended the non-Newtonian calculus to the complex valued functions and was interested in the statements of some fundamental