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Some paranormed sequence spaces of non-absolute type derived by weighted mean. (English) Zbl 1105.46005

Summary: The sequence spaces \(\ell_\infty(p)\), \(c(p)\) and \(c_0(p)\) were introduced and studied by I. J.Maddox, Proc.Camb.Philos.Soc.64, 335–340 (1968; Zbl 0157.43503)]. In the present paper, the sequence spaces \(\lambda(u,v;p)\) of non-absolute type which are derived by the generalized weighted mean are defined and it is proved that the spaces \(\lambda(u,v;p)\) and \(\lambda(p)\) are linearly isomorphic, where \(\lambda\) denotes one of the sequence spaces \(\ell_\infty\), \(c\) or \(c_0\). Besides this, the \(\beta\)- and \(\gamma\)-duals of the spaces \(\lambda(u,v;p)\) are computed and basis of the spaces \(c_0(u,v;p)\) and \(c(u,v;p)\) is constructed. Additionally, it is established that the sequence space \(c_0(u,v)\) has the AD property and the \(f\)-dual of the space \(c_0(u,v;p)\) is given. Finally, the matrix mappings from the sequence spaces \(\lambda(u,v;p)\) to the sequence space \(\mu\) and from the sequence space \(\mu\) to the sequence spaces \(\lambda(u,v;p)\) are characterized.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0157.43503
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References:

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