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Some new paranormed sequence spaces. (English) Zbl 1049.46002

Summary: I. J. Maddox defined the sequence spaces \(\ell_{\infty} (p), c(p)\) and \(c_0(p)\) in [Proc. Camb. Philos. Soc. 64, 335–340 (1968; Zbl 0157.43503), Q. J. Math., Oxf. (2) 18, 345–355 (1967; Zbl 0156.06602)]. In the present paper, the sequence spaces \(a_0^r(u,p)\) and \(a_c^r(u,p)\) of non-absolute type are introduced and it is proved that the spaces \(a_0^r(u,p)\) and \(a_c^r(u,p)\) are linearly isomorphic to the spaces \(c_0(p)\) and \(c(p)\), respectively. Besides this, the \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals of the spaces \(a_0^r(u,p)\) and \(a_c^r(u,p)\) are computed and their bases are constructed. Finally, a basic theorem is given and some matrix mappings from \(a_0^r(u,p)\) to the sequence spaces of Maddox and to new sequence spaces are characterized.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
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References:

[1] Ahmad, Z. U.; Mursaleen, Köthe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd), 42, 56, 57-61 (1987) · Zbl 0647.46006
[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] B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 27 (to appear); B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 27 (to appear)
[4] C. Aydın, F. Başar, On the new sequence spaces which include the spaces \(c_0c\); C. Aydın, F. Başar, On the new sequence spaces which include the spaces \(c_0c\)
[5] Başar, F., Infinite matrices and almost boundedness, Boll. Un. Mat. Ital. A, 6, 7, 395-402 (1992), (no. 3) · Zbl 0867.47021
[6] Başar, F.; Altay, B., Matrix mappings on the space \(bs (p)\) and its \(α-, β\)- and \(γ\)-duals, Aligarh Bull. Math., 21, 1, 79-91 (2002)
[7] F. Başar, B. Altay, Mursaleen, Some generalizations of the space \(bv_pp\); F. Başar, B. Altay, Mursaleen, Some generalizations of the space \(bv_pp\)
[8] Başarır, M., On some new sequence spaces and related matrix transformations, Indian J. Pure Appl. Math., 26, 10, 1003-1010 (1995) · Zbl 0855.40005
[9] Choudhary, B.; Mishra, S. K., On Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math., 24, 5, 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] Kampthan, P. K.; Gupta, M., Sequence Spaces and Series (1981), Marcel Dekker Inc: Marcel Dekker Inc New York, Basel
[12] Maddox, I. J., Elements of Functional Analysis (1988), The University Press: The University Press Cambridge · Zbl 0193.08601
[13] Maddox, I. J., Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Philos. Soc., 64, 335-340 (1968) · Zbl 0157.43503
[14] Maddox, I. J., Spaces of strongly summable sequences, Quart. J. Math. Oxford, 18, 2, 345-355 (1967) · Zbl 0156.06602
[15] Mursaleen, F. Başar, B. Altay, On the Euler sequence spaces which include the spaces \(ℓ_p_∞\); Mursaleen, F. Başar, B. Altay, On the Euler sequence spaces which include the spaces \(ℓ_p_∞\)
[16] Nakano, H., Modulared sequence spaces, Proc. Jpn. Acad., 27, 2, 508-512 (1951) · Zbl 0044.11302
[17] Ng, P.-N.; Lee, P.-Y., Cesàro sequences spaces of non-absolute type, Comment Math. Prace Mat., 20, 2, 429-433 (1978) · Zbl 0408.46012
[18] Simons, S., The sequence spaces ℓ \((p_v)\) and \(m(p_v)\), Proc. London Math. Soc., 15, 3, 422-436 (1965) · Zbl 0128.33805
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