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Generalization of the sequence space \(\ell (p)\) derived by weighted mean. (English) Zbl 1116.46003

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.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)

Citations:

Zbl 0156.06602
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References:

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