Altay, Bilâl; Başar, Feyzi Generalization of the sequence space \(\ell (p)\) derived by weighted mean. (English) Zbl 1116.46003 J. Math. Anal. Appl. 330, No. 1, 174-185 (2007). Summary: The sequence space \(\ell(p)\) was introduced and studied by I.J.Maddox [Q. J. Math., Oxf., II.Ser.18, 345–355 (1967; Zbl 0156.06602)]. In the present paper, the sequence spaces \(\ell(u,v;p)\) of non-absolute type which are derived by the generalized weighted mean are defined and it is proved that the spaces \(\ell(u,v;p)\) and \(\ell(p)\) are linearly isomorphic. Besides this, the \(\beta\)- and \(\gamma\)-duals of the space \(\ell(u,v;p)\) are computed and a basis of that space is constructed. Further, it is established that the sequence space \(\ell_p(u,v)\) has the AD property and the \(f\)-dual of the space \(\ell_p (u,v)\) is given. Finally, the matrix mappings from the sequence spaces \(\ell(u,v;p)\) to the sequence space \(\mu\) and from the sequence space \(\mu\) to the sequence space \(\ell(u,v;p)\) are characterized. Cited in 2 ReviewsCited in 69 Documents MSC: 46A45 Sequence spaces (including Köthe sequence spaces) Keywords:paranormed sequence space; \(f\)-, \(\beta\)-, and \(\gamma\)-duals; weighted mean; AD property and matrix mappings Citations:Zbl 0156.06602 PDFBibTeX XMLCite \textit{B. Altay} and \textit{F. Başar}, J. Math. Anal. Appl. 330, No. 1, 174--185 (2007; Zbl 1116.46003) Full Text: DOI References: [1] Altay, B.; Başar, F., Some new spaces of double sequences, J. Math. Anal. Appl., 309, 70-90 (2005) · Zbl 1093.46004 [2] Altay, B.; Başar, F., On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 26, 701-715 (2002) · Zbl 1058.46002 [3] Altay, B.; Başar, F., Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Anal. Appl., 319, 494-508 (2006) · Zbl 1105.46005 [4] Aydın, C.; Başar, F., Some new sequence spaces which include the spaces \(\ell_p\) and \(\ell_\infty \), Demonstratio Math., 38, 641-656 (2005) · Zbl 1096.46005 [5] Başar, F., Infinite matrices and almost boundedness, Boll. Unione Mat. Ital. Ser. A (7), 6, 395-402 (1992) · Zbl 0867.47021 [6] Başar, F.; Altay, B., Matrix mappings on the space \(b s(p)\) and its \(α-, β\)- and \(γ\)-duals, Aligarh Bull. Math., 21, 79-91 (2001) [7] Başar, F.; Altay, B., On the space of sequences of \(p\)-bounded variation and related matrix mappings, Ukrainian Math. J., 55, 136-147 (2003) · Zbl 1040.46022 [8] Çolak, R.; Et, M.; Malkowsky, E., Some topics of sequence spaces, (Lecture Notes in Math. (2004), Fırat Univ. Elâzığ: Fırat Univ. Elâzığ Turkey), 1-63 [9] Choudhary, B.; Mishra, S. K., On Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math., 24, 291-301 (1993) · Zbl 0805.46008 [10] Grosse-Erdmann, K.-G., Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl., 180, 223-238 (1993) · Zbl 0791.47029 [11] Maddox, I. J., Spaces of strongly summable sequences, Quart. J. Math. Oxford (2), 18, 345-355 (1967) · Zbl 0156.06602 [12] Maddox, I. J., Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc., 64, 335-340 (1968) · Zbl 0157.43503 [13] Maddox, I. J., Elements of Functional Analysis (1988), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK · Zbl 0193.08601 [14] Malkowsky, E., Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik, 49, 187-196 (1997) · Zbl 0942.40006 [15] Malkowsky, E.; Savaş, E., Matrix transformations between sequence spaces of generalized weighted means, Appl. Math. Comput., 147, 333-345 (2004) · Zbl 1036.46001 [16] Nakano, H., Modulared sequence spaces, Proc. Japan Acad., 27, 508-512 (1951) · Zbl 0044.11302 [17] Ng, P.-N.; Lee, P.-Y., Cesàro sequence spaces of non-absolute type, Comment. Math. Prace Mat., 20, 429-433 (1978) · Zbl 0408.46012 [18] Simons, S., The sequence spaces \(\ell(p_v)\) and \(m(p_v)\), Proc. London Math. Soc. (3), 15, 422-436 (1965) · Zbl 0128.33805 [19] Wilansky, A., Summability through Functional Analysis, Mathematics Studies, vol. 85 (1984), North-Holland: North-Holland Amsterdam · Zbl 0531.40008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.