# ON THE DUAL SPACE OF CONE NORMED SPACES

## Abstract

Cone normed spaces are the generalization of the normed spaces with many authors adjusting the theory to the classical one. Despite all the efforts of researchers in generalizing the theory, there is no specific research on the duality of cone normed space in the literature. In this paper, we investigate and study some properties of the space of all continuous linear mappings between cone normed spaces, this allows us to define the concept of dual in the setting of cone normed spaces, state some of its properties and used the properties to prove the Hahn-Banach Theorem in cone normed space.

Keywords:
Continuous linear map, cone norm, semi-cone norm, dual space, Hahn-Banach theorem.

## Article Details

How to Cite
YUSUF, A., GARBA, A. I., & NAKONE, B. (2020). ON THE DUAL SPACE OF CONE NORMED SPACES. Asian Journal of Mathematics and Computer Research, 27(1), 38-45. Retrieved from https://www.ikprress.org/index.php/AJOMCOR/article/view/5017
Section
Original Research Article

## References

réchet MM. Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo (1884-1940). 1906;22(1):1–72.

Banach S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 1922;3(1):133–181.

Hahn H. Über folgen linearer operationen. Monatshefte für Mathematik. 1922;32(1):3–88.

Wiener N. Limit in terms of continuous transformation. Bulletin de la Société Mathématique de France. 1922;50:119–134.

Rzepecki B. On fixed point theorems of Maia type, Publications de l’Institut Mathématique. 1980;28(42): 179–186.

Samanta T, Roy S, Dinda B. Cone normed linear spaces. arXiv preprint arXiv:1009.2172; 2010.

Sonmez A, Cakalli H. Cone normed spaces and weighted means. Mathematical and Computer Modelling. 2010;52(9-10):1660–1666.
Available:https://doi.org/10.1016/j.mcm.2010.06.032

Abdeljawad T, Turkoglu D, Abuloha M. Some theorems and examples of cone Banach spaces. Journal of Computational Analysis and Applications. 2010;12(4):739–753.

Eshaghi GM, Ramezani M, Khodaei H, Baghani H. Cone normed spaces. Caspian Journal of Mathematical Sciences (CJMS). 2012;1(1).

Tamang P, Bag T. Some fixed point results in fuzzy cone normed linear space. Journal of the Egyptian Mathematical Society. 2019;27(1):46.
Available:https://doi.org/10.1186/s42787-019-0046-6.

İlkhan M, Alp PZ, Kara EE. On the spaces of linear operators acting between asymmetric cone normed spaces, Mediterranean Journal of Mathematics. 2018;15(3):136.
Available:https://doi.org/10.1007/s00009-018-1182-0

Sarkar K, Tiwary K. Fixed point theorem in cone banach spaces. International Journal of Statistics and Applied Mathematics. 2018;3(4):143–146.

Chen GY, Huang X, Yang X. Vector optimization: Set-valued and variational analysis. Springer Science & Business Media. 2006;541.

Çakallı H, Sönmez A, Genç Ç. On an equivalence of topological vector space valued cone metric spaces and metric spaces, Applied Mathematics Letters. 2012;25(3):429–433.
Available:https://doi.org/10.1016/j.aml.2011.09.029.

Kreyszig E. Introductory functional analysis with applications, Wiley New York; 1978.