Review on Some Sequence Spaces of p-adic Numbers

Authors : Orhan Tuğ1 & Mutlay Doğan2
1&2Mathematics Education Department, Ishik University, Erbil, Iraq

Abstract:   In this paper, we make a literature review on p-adic sequences and we prove some new topological properties on the sequence spaces w(p), l(p) and c(p) of p-adic numbers as the set of all sequences, the set of bounded sequences and the set of convergent sequences of p-adic numbers, respectively. We show that these sequence spaces are Banach spaces under some certain topological properties. Moreover we prove some inclusion relation between these sequence spaces.We construct the α-, β- and γ duals of sequence spaces w(p), l(p) and c(p), and of p-adic numbers. We conclude the paper with characterizations of some significant matrix classes.

Keywords: Sequence Spaces, P-Adic Numbers, Banach Space, P-Adic Sequences, Matrix Transformations


doi: 10.23918/eajse.v3i1sip231


Download the PDF Document from here.


References
Andree, R. V., & Petersen, G. M. (1956). Matrix methods of summation, regular for -adic
valuations. Proceedings of the American Mathematical Society, 7(2), 250-253.
Bachman, G. (1964). Introduction to p-adic Numbers and Valuation Theory. Academic Press. Inc.
Borwein, D., & Jakimovski, A. (1994). Matrix transformations of power series. Proceedings of the
American Mathematical Society, 122(2), 511-523.
Cho, I. (2014). p-adic Banach space operators and adelic Banach space operators. Opuscula
Mathematica, 34.
Gouvêa, F. Q. (1997). Elementary Analysis in p. In p-adic Numbers (pp. 87-132). Springer Berlin
Heidelberg.
Katsaras, A. K. (1990). On non-Archimedean sequence spaces. Bull. Inst. Math. Acad. Sinica, 18(2).
Katok, S. (2007). p-adic Analysis Compared with Real (Vol. 37). American Mathematical Soc.,
University Park PA, USA.
Khrennikov, A. (1997). Non-archimedean analysis. In Non-Archimedean Analysis: Quantum
Paradoxes, Dynamical Systems and Biological Models (pp. 101-129). Springer
Netherlands.
Koblitz, N. (1977). P-adic numbers. In p-adic Numbers, p-adic Analysis, and Zeta-Functions (pp. 1-
20). Springer US.Mahler, K. (1973). Introduction to P-Adic Numbers and Their Function.
CUP Archive.
Monna, A. F. (1970). Analyse non-archimedienne.
Natarajan, P. N. (2012). Some characterizations of Schur matrices in ultrametric fields. Comment.
Math. Prace Mat, 52, 137-142.
Parent, D. P. (1984). p-Adic Analysis. In Exercises in Number Theory (pp. 466-532). Springer New
York.
Robert, A. M. (2000). A course in p-adic analysis, volume 198 of Graduate Texts in Mathematics.
Sally, P. J. (1998). An introduction to p-adic fields, harmonic analysis and the representation theory
of SL2. Letters in Mathematical Physics, 46(1), 1-47.