Some new Pascal sequence spaces

Abstract

The main purpose of the present paper is to study of some new Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$. New Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$ are found as $BK$-spaces and it is proved that the spaces $% p_{\infty }$, $p_{c}$ and $p_{0}~$are linearly isomorphic to the spaces $% l_{\infty }$, $c\ $and $c_{0}$ respectively. Afterward, $\alpha $-, $\beta $% -~and $\gamma $-duals of these spaces $p_{c}$ and $p_{0}$ are computed and their bases are consructed. Finally, matrix the classes $(p_{c}:l_{p})$ and $% (p_{c}:c)$ have been characterized.

Keywords

• $\alpha $-,$\beta $-~and $\gamma $-duals and basis of sequence,Matrix mappings,Pascal sequence spaces

References

1. [1] B. Choudhary and S. Nanda, Functional Analysis with Applications, Wiley, New Delhi, 1989.
2. [2] M. Kirisçi, On the Taylor Sequence Spaces of Non-Absolute Type which Include The Spaces c0 and c, Journal of Math. Analysis 6 (2015), 22-35.
3. [3] B. Altay, F. Başar, Some Euler Sequence Spaces of Non-Absolute Type, Ukrainian Math. J. 57 (2005), 1-17.
4. [4] M. Şengönül, F. Başar, Some New Cesaro Sequence Spaces of Non-Absolute Type which Include The Spaces c0 and c, Soochow J. Math. 31 (2005), 107-119.
5. [5] B. Altay, F. Başar, M. Mursaleen, On the Euler sequence spaces which include in the spaces lp and l, Inform. Sci. 176 (2006), 1450-1462.
6. [6] E. Malkowsky, Recent results in the theory of matrix transformations in sequences spaces, Mat. Vesnik 49 (1997), 187-196.
7. [7] P. N. Ng, P. Y. Lee, Cesaro sequences spaces of non-absolute type, Comment. Math. Prace Mat. 20 (1978), 429-433.
8. [8] C. S. Wang, On Norlund seqence spaces, Tamkang J. Math. 9 (1978), 269-274.

9. [9] G. H. Lawden, Pascal matrices, Mathematical Gazette 56 (1972), 325-327.
10. [10] S. Dutta and P. Baliarsingh, On some Toeplitz matrices and their inversions, J. Egypt Math. Soc. 22 (2014), 420-423.
11. [11] R. Brawer, Potenzen der Pascal matrix und eine Identitat der Kombinatorik, Elem. der Math. 45 (1990), 107-110.
12. [12] A. Edelman and G. Strang, Pascal Matrices, The Mathematical Association of America, Monthly 111 (2004), 189-197.
13. [13] C. Lay David, Linear Algebra and Its Applications, 4th Ed. Boston, Pearson, Addison-Wesley, 2012.
14. [14] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, Cambridge, 1981.
15. [15] D.J.H. Garling, The α, β and γ Duality of Sequence Spaces, Proc. Comb. Phil. Soc. 63 (1967), 963-981.
16. [16] M. Stieglitz, H. Tietz, Matrixtransformationen von Folgenraumen Eine Ergebnis¨ubersict, Math. Z. 154 (1977), 1-16.