Abstract
An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.
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References
Abdeljawad, T.: Completion of cone metric spaces. Hacet. J. Math. Stat. 39(1), 67–74 (2010)
Abdeljawad, T., Türkoğlu, D., Abuloha, M.: Some theorems and examples of cone Banach spaces. J. Comput. Anal. Appl. 12(4), 739–753 (2010)
Alegre, C.: Continuous operators on asymmetric normed spaces. Acta. Math. Hungar. 122(4), 357–372 (2009)
Alegre, C., Ferrando, I., Garcia-Raffi, L.M., Sanchez-Perez, E.A.: Compactness in asymmetric normed spaces. Topology Appl. 155(6), 527–539 (2008)
Cobzas, S.: Functional analysis in asymmetric normed spaces. Birkhauser, Basel (2012)
Collins, J., Zimmer, J.: An asymmetric Arzela Ascoli theorem. Topol. Appl. 154, 2312–2322 (2007)
Deimling, K.: Nonlinear functional analysis. Springer, Berlin (1985)
Garcia-Raffi, L.M.: Compactness and finite dimension in asymmetric normed linear spaces. Topol. Appl. 153, 844–853 (2005)
Garcia-Raffi, L.M., Romaguera, S., Sanchez-Perez, E.A.: Sequence spaces and asymmetric norms in the theory of computational complexity. Math. Comput. Model. 36, 1–11 (2002)
Garcia-Raffi, L.M., Romaguera, S., Sanchez-Perez, E.A.: The bicompletion of an asymmetric normed linear space. Acta Math. Hungar. 97, 183–191 (2002)
Garcia-Raffi, L.M., Romaguera, S., Sanchez-Perez, E.A.: The dual space of an asymmetric normed linear space. Quaest. Math. 26(1), 83–96 (2003)
Gordji, M.E., Ramezani, M., Khodaei, H., Baghani, H.: Cone normed spaces. Caspian J. Math. Sci. 1(1), 7–12 (2012)
İlkhan, M., Kara, E.E.: On some results related to asymmetric cone normed spaces (submitted)
Kelly, J.C.: Bitopological spaces. J. Proc. Lond. Math. Soc. 13, 71–89 (1963)
Long-Guang, H., Xian, Z.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, 1468–1476 (2007)
Radenovic, S., Kadelburg, Z.: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5, 38–50 (2011)
Reilly, I.L., Subrahmanyam, P.V., Vamanamurthy, M.K.: Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte für Mathematik 93(127), 127–140 (1982)
Reilly, I.L., Vamanamurthy, M.K.: On oriented metric spaces. Math. Slovaca 34(3), 299–305 (1984)
Romaguera, S.: Left \(K\)-completeness in quasi metric spaces. Math. Nachr. 157, 15–23 (1992)
Romaguera, S., Gutierrez, A.: A note on Cauchy sequences in quasi pseudo metric spaces, Glas. Mat. Ser. III 21(41):1, 191-200 (1986)
Shaddad, F., Md Noorani, M.S.: Fixed point results in quasi-cone metric spaces, Abstr. Appl. Anal., Article ID 303626, 7 pages (2013)
Sönmez, A.: On paracompactness in cone metric spaces. Appl. Math. Lett. 23, 494–497 (2010)
Sönmez, A., Cakallı, H.: Cone normed spaces and weighted means. Math. Comput. Model. 52, 1660–1666 (2010)
Türkoğlu, D., Abuloha, M.: Cone metric spaces and fixed point theorems in diametrically contractive mappings. Acta Math. Sin. (Engl. Ser.) 26(3), 489–496 (2010)
Wilson, A.W.: On quasi metric spaces. Am. J. Math. 53, 675–684 (1931)
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İlkhan, M., Zengin Alp, P. & Kara, E.E. On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces. Mediterr. J. Math. 15, 136 (2018). https://doi.org/10.1007/s00009-018-1182-0
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DOI: https://doi.org/10.1007/s00009-018-1182-0