Consistency Of Definition Of Orthogonality In A Partial Metric Induced By A Metric
DOI:
https://doi.org/10.31316/jderivat.v10i1.4620Abstract
An extension of the metric space in which the distance of the same point is not always zero is called a partial metric space. Orthogonality is the relation of two perpendicular lines at one point of intersection forming a right angle. There are several ways to define orthogonality, including Pythagorean Orthogonality, Isosceles Orthogonality, and Birkhoff-James Orthogonality. The purpose of this research is to study the consistency of the definition of orthogonality in the metric space to the partial metric space. Based on these results, it can be concluded that the partial metric space can be obtained by linear induction from a metric space. Then, in developing the definition of orthogonality to the partial metric space, it can be concluded that the qualified orthogonality is the I-orthogonality and the BJ-orthogonality, while the P-orthogonality does not qualify the consistency of the definition of orthogonality in the partial metric space.
Kata Kunci: Orthogonality, Consistency, and Partial metric space
References
Anton, H. (2011). Elementary Linear Algebra: Aplicatons Version (Tenth Edit). New York: John Wiley and Sond.
Bugajewski, D., & Wang, R. (2020). On the topology of partial metric spaces. Mathematica Slovaca, 70(1), 135–146. https://doi.org/10.1515/ms-2017-0338
Bukatin, M., Kopperman, R., Matthews, S., & Pajoohesh, H. (2009). Partial Metric Spaces. American Mathematical Monthly, 116(8), 708–718. https://doi.org/10.4169/193009709X460831
Chmieliński, J., & Wójcik, P. (2010). On a ρ-orthogonality. Aequationes Mathematicae, 80(1), 45–55. https://doi.org/10.1007/s00010-010-0042-1
Chmieliński, J., & Wójcik, P. (2018). Approximate symmetry of Birkhoff orthogonality. Journal of Mathematical Analysis and Applications, 461(1), 625–640.
Diminnie, C. R. (1983). A New Orthogonality Relation for Normed Linear Spaces. Mathematische Nachrichten, 114(1), 197–203. https://doi.org/10.1002/mana.19831140115
Frechet, M. (1906). Sur Quelques Points Du Calcul Fonctionnel. Monatshefte Fur Mathematik Und Physik, 19(1), 47–48.
Gusnedi. (1999). Matriks dan Ruang Vektor. DIP Universitas Negri Padang.
Han, S., Wu, J., & Zhang, D. (2017). Properties and principles on partial metric spaces. Topology and Its Applications, 44(3), 77–98. https://doi.org/10.1016/j.topol.2017.08.006
Javed, K., Aydi, H., Uddin, F., & Arshad, M. (2021). On Orthogonal Partial b-Metric Spaces with an Application. Journal of Mathematics, 2021(3), 1–7. https://doi.org/10.1155/2021/6692063
Joseph A Gallian. (2017). Contemporary Abstract Algebra (ninth edit). Cengage Learning.
Kir Mehmet, K. H. (2016). Generalized Fixed Point Theorems in Partial Metric Space. European Journal Of Pure And Applied Mathematics, 9(4), 443–451. https://doi.org/10.11568/kjm.2012.20.3.353
Matthews. S.G. (1994). Partial Metric Topology. Topology and Its Applications, 48(3), 183–197.
Mykhaylyuk, V., & Myronyk, V. (2019). Compactness and completeness in partial metric spaces. Topology and Its Applications, 44(3), 1–10. https://doi.org/10.1016/j.topol.2019.106925
Partington, J. R. (1986a). Orthogonality in normed spaces. Bulletin of the Australian Mathematical Society, 33(3), 449–455. https://doi.org/10.1017/S0004972700004020
Partington, J. R. (1986b). Orthogonality in normed spaces. Bulletin of the Australian Mathematical Society, 33(3), 449–455. https://doi.org/10.1017/S0004972700004020
Senapati, T. (2018). Weak orthogonal metric spaces and fixed point results. Kragujevac Journal Of Mathematics, 42(4), 1–12.
Tawfeek, S., Faried, N., & El-Sharkawy, H. A. (2021). Orthogonality in smooth countably normed spaces. Journal of Inequalities and Applications, 50(1), 20. https://doi.org/10.1186/s13660-020-02531-5
Uddin, F., Park, C., Javed, K., Arshad, M., & Lee, J. R. (2021). Orthogonal m-metric spaces and an application to solve integral equations. Advances in Difference Equations, 51(2), 159. https://doi.org/10.1186/s13662-021-03323-x
William F. Ames. (2014). Numerical Methods for Partial Differential Equations (Werner Rheinboldt, Ed.; Third Edit). Academic Press.
Downloads
Published
Issue
Section
Citation Check
License
Copyright (c) 2023 Mochammad Hafiizh, Nila Puspita Dewi, Sisworo, Dahliatul Hasanah
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Authors who publish with this journal agree to the following terms:
-
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution-ShareAlike 4.0 International License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
