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International Journal of Analysis
Volume 2014 (2014), Article ID 746059, 10 pages
Research Article

The Construction of Hilbert Spaces over the Non-Newtonian Field

1Department of Mathematics, Faculty of Sciences, Gazi University, 06500 Ankara, Turkey
2Department of Mathematics, Faculty of Sciences and Arts, Bozok University, 66100 Yozgat, Turkey

Received 23 January 2014; Revised 20 May 2014; Accepted 3 June 2014; Published 21 October 2014

Academic Editor: Seenith Sivasundaram

Copyright © 2014 Uğur Kadak and Hakan Efe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.