**Authors** : Orhan Tuğ^{1} & Mutlay Doğan^{2}

^{1&2}Mathematics 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

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**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.