×

On the paranormed Nörlund sequence space of nonabsolute type. (English) Zbl 1468.46014

Summary: I. J. Maddox [Q. J. Math., Oxf. II. Ser. 18, 345–355 (1967; Zbl 0156.06602)] defined the space \(\ell(p)\) of the sequences \(x = (x_k)\) such that \(\sum_{k = 0}^{\infty} | x_k |^{p_k} < \infty\). In the present paper, the Nörlund sequence space \(N^t(p)\) of nonabsolute type is introduced and proved that the spaces \(N^t(p)\) and \(\ell(p)\) are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the space \(N^t(p)\) are computed and the basis of the space \(N^t(p)\) is constructed. The classes \((N^t(p) : \mu)\) and \((\mu : N^t(p))\) of infinite matrices are characterized. Finally, some geometric properties of the space \(N^t(p)\) are investigated.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0156.06602
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Maddox, I. J., Spaces of strongly summable sequences, Quarterly Journal of Mathematics, 18, 1, 345-355 (1967) · Zbl 0156.06602 · doi:10.1093/qmath/18.1.345
[2] Nakano, H., Modulared sequence spaces, Proceedings of the Japan Academy, 27, 2, 508-512 (1951) · Zbl 0044.11302 · doi:10.3792/pja/1195571225
[3] Simons, S., The sequence spaces \(\ell(p_\upsilon)\) and \(m(p_\upsilon)\), Proceedings of the London Mathematical Society, 15, 3, 422-436 (1965) · Zbl 0128.33805
[4] Peyerimhoff, A., Lectures on Summability. Lectures on Summability, Lecture Notes in Mathematics (1969), New York, NY, USA: Springer, New York, NY, USA · Zbl 0182.08401
[5] Mears, F. M., The inverse Nörlund mean, Annals of Mathematics, 44, 3, 401-409 (1943) · Zbl 0061.12202 · doi:10.2307/1968971
[6] Choudhary, B.; Mishra, S. K., On Köthe-Toeplitz duals of certain sequence spaces and thair matrix transformations, Indian Journal of Pure and Applied Mathematics, 24, 59, 291-301 (1993) · Zbl 0805.46008
[7] Başar, F.; Altay, B., Matrix mappings on the space bs(p) and its \(\alpha-, \beta \)- and \(\gamma \)-duals, The Aligarh Bulletin of Mathematics, 21, 1, 79-91 (2002)
[8] Başar, F., Infinite matrices and almost boundedness, Bollettino della Unione Matematica Italiana A, 6, 7, 395-402 (1992) · Zbl 0867.47021
[9] Altay, B.; Başar, F., On the paranormed Riesz sequence space of non-absolute type, Southeast Asian Bulletin of Mathematics, 26, 701-715 (2002) · Zbl 1058.46002
[10] Wang, C. S., On Nörlund sequence space, Tamkang Journal of Mathematics, 9, 269-274 (1978) · Zbl 0415.46009
[11] Grosse-Erdmann, K. G., Matrix transformations between the sequence spaces of Maddox, Journal of Mathematical Analysis and Applications, 180, 1, 223-238 (1993) · Zbl 0791.47029 · doi:10.1006/jmaa.1993.1398
[12] Lascarides, C. G.; Maddox, I. J., Matrix transformations between some classes of sequences, Proceedings of the Cambridge Philosophical Society, 68, 99-104 (1970) · Zbl 0193.41102 · doi:10.1017/S0305004100001109
[13] Chen, S., Geometry of Orlicz spaces, Dissertationes Mathematicae, 356, 1-224 (1996) · Zbl 1089.46500
[14] Diestel, J., Geomety of Banach Spaces-Selected Topics (1984), Berlin, Germany: Springer, Berlin, Germany
[15] Maligranda, L., Orlicz Spaces and Interpolation (1985), Poznan, Poland: Institute of Mathematics Polish Academy of Sciences, Poznan, Poland
[16] Nergiz, H.; Başar, F., Some geometric properties of the domain of the double sequential band matrix \(B(\widetilde{r}, \widetilde{s})\) in the sequence space \(\ell(p)\), Abstract and Applied Analysis, 2013 (2013) · Zbl 1282.46020 · doi:10.1155/2013/421031
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.