On the Algebra of Interval Vectors
Abstract
In this study, we examine some important subspaces by showing that the set of n-dimensional interval vectors is a quasilinear space. By defining the concept of dimensions in these spaces, we show that the set of $n$-dimensional interval vectors is actually a $(n_{r},n_{s})$-dimensional quasilinear space and any quasilinear space is $\left( n_{r},0_{s}\right) $-dimensional if and only if it is $n$-dimensional linear space. We also give examples of $(2_{r},0_{s})$ and $(0_{r},2_{s})$-dimensional subspaces. We define the concept of dimension in a quasilinear space with natural number pairs. Further, we define an inner product on some spaces and talk about them as inner product quasilinear spaces. Further, we show that some of them have Hilbert quasilinear space structure.
Keywords
Quasilinear space, Interval vectors, Inner product quasilinear space, Hilbert quasilinear space
References
- [1] Aseev, S.M.: Quasilinear operators and their application in the theory of multivalued mappings. Proceedings of the Steklov Institute of Mathematics. Issue 2, 23-52 (1986).
- [2] Moore, E. R., Kearfott, R. B., Cloud, M. J.: Introduction to Interval Analysis. SIAM. Philadelphia, (2009).
- [3] Banazılı, H.K.: On quasilinear operators between quasilinear spaces. İnönü University, M.Sc. Thesis, Malatya, (2014).
- [4] Bozkurt, H.: Quasilinear inner product spaces and some generalizations. İnönü University, PhD Thesis, Malatya, (2016).
- [5] Yılmaz, Y., Çakan S., Aytekin, ¸S.: Topological Quasilinear Spaces. Abstract and Applied Analysis. Article ID 951374, 10 pages (2012).
- [6] Çakan S., Yılmaz, Y.: Normed proper quasilinearmspaces. Journal of Nonlinear Science and Applications. 8, 816-836 (2015).
- [7] Çakan S., Yılmaz, Y.: Riesz Lemma in Normed Quasilinear Spaces and Its an Application. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 88, (2), 231-239 (2018).
- [8] Bozkurt, H., Yılmaz, Y.: Some new results on inner product quasilinear spaces. Cogents Mathematics3, 1194801 (2016).
- [9] Yılmaz, Y., Bozkurt, H., Çakan S.: On ortonormal sets in inner product quasilinear spaces. Creative Mathematics and Informatics. 25, 229-239 (2016).
- [10] Bozkurt, H., Yılmaz, Y.: New Inner Product Quasilinear Spaces on Interval Numbers. Journal of Function Spaces. Volume 2016, Article ID 2619271, 9 pages (2016).