SOME PROPERTIES OF SEQUENCE SPACE _ BV (f; p; q; s)

Year 2018, Volume: 67 Issue: 1, 235 - 241, 01.02.2018

Mahmut Işık

https://doi.org/10.1501/Commua1_0000000845

Cited By: 2

Abstract

Let `1 and c denote the Banach spaces of real bounded and convergent sequences
x = (xn) normed by kxk = sup
n
jxnj ; respectively.
Let be a one to one mapping of the set of positive integers into itself such that

k
(n) =


k1
(n)

; k = 1; 2; ::: .A continuous linear functional ' on `1 is said
to be an invariant mean or a mean if and only if
(i) ' (x) 0 when xn 0 for all n;
(ii) ' (e) = 1; where e = (1; 1; 1; :::) and
(iii) '
x(n)
= ' (fxng) for all x 2 `1:
If is the translation mapping n ! n + 1; a mean is often called a Banach
limit [3], and V is the set of convergent sequences, that is, the set of bounded
sequences all of whose invariant means are equal, is the set ^f of almost convergent
sequences

Keywords

Modulus function, sequence spaces, seminorm

References

• Ahmad, Z.U. and Mursaleen, M. An application of Banach limits, Proc. Amer. Math. Soc. , (1988), 244-246.
• Altinok, H. Altin, Y. I¸sik, M. The sequence space BV (M; p; q; s) on seminormed spaces. Indian J. Pure Appl. Math. 39(1) (2008), 49–58
• Banach, S. Theorie des Operations Linearies, Warszawa, 1932.
• Bhardwaj, V.K. A generalization of a sequence space of Ruckle, Bull. Calcutta Math. Soc. (5) (2003), 411-420.
• Et, M. Spaces of Cesàro diğerence sequences of order r de…ned by a modulus function in a locally convex space. Taiwanese J. Math. 10(4) (2006), 865–879.
• Et, M. : Strongly almost summable diğerence sequences of order m de…ned by a modulus. Studia Sci. Math. Hungar. 40(4) (2003), 463–476.
• Freedman, A.R. Sember, J. J. Raphael, M. Some Cesàro-type summability spaces. Proc. London Math. Soc. 3(3) 37 (1978), 508–520.
• I¸sik, M. Generalized vector-valued sequence spaces de…ned by modulus functions. J. Inequal. Appl. 2010, Art. ID 457892, 7 pp.
• I¸sik, M. Strongly almost (w; ; q) summable sequences. Math. Slovaca. 61(5) (2011), 779–
• Karakaya, V. and Sava¸s, E. On almost p bounded variation of lacunary sequences. Comput. Math. Appl. 61(6) (2011), 1502–1506.
• Lorentz, G. G. A contribution the theory of divergent series, Acta Math. 80 (1948), 167-190.
• Maddox.I. J. Sequence spaces de…ned by a modulus, Math. Proc. Camb. Phil. Soc. 100 (1986), 166.
• Mohiuddine, S. A. An application of almost convergence in approximation theorems. Appl. Math. Lett. 24 (2011), no. 11, 1856–1860
• Mohiuddine, S. A. Matrix transformations of paranormed sequence spaces through de la Vallee-Pousion mean, Acta Scientiarum,Technology,37(1) (2015),71-75.
• Mursaleen, M. Mohiuddine, S. A. Some matrix transformations of convex and paranormed sequence spaces into the spaces of invariant means. J. Funct. Spaces Appl. 2012, Art. ID , 10 pp
• Mursaleen, M. On some new invariant matrix methods of summability, Quart. J. Math. Oxford 34(2), (1983), 77-86.
• Mursaleen, M. Matrix transformations between some new sequence spaces, Houston J. Math. , (1983), 505-509.
• Nakano,H. Concave modulars, J. Math. Soc. Japan. 5 (1953), 29-49.
• Nanda, S. and Nayak, K. C. Some new sequence spaces, Indian J.Pure Appl.Math. 9(8) (1978) 846.
• Raimi, R. A. Invariant means and invariant matrix method of summability, Duke Math. J. , (1963), 81-94.
• Ruckle,W. H. FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math. 25 (1973), 973-978.
• Sava¸s, E. and Patterson, R. F. Double sequence spaces de…ned by a modulus. Math. Slovaca (2) (2011), 245–256.
• Sava¸s, E. On some new double sequence spaces de…ned by a modulus. Appl. Math. Comput. (1) (2007), 417–424.
• Schaefer, P. In…nite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104
• Wilansky, A. Functional Analysis, Blaisdell Publishing Company, New York, 1964.
• Current address : Harran University, Faculty of Education, Sanliurfa-TURKEY E-mail address : [e-mail]