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The sequence spaces ∞ (p), c(p) and c 0 (p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u, v; p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ (u, v; p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ∞ , c or c 0 . Besides this, the β-and γ -duals of the spaces λ (u, v; p) are computed and the 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 AD property and given the f -dual of the space c 0 (u, v; p). Finally, the matrix mappings from the sequence spaces λ (u, v; p) to the sequence space μ and from the sequence space μ to the sequence spaces λ (u, v; p) are characterized.

517.9 In the present paper, we introduce Euler sequence spaces e r 0 and e c r of nonabsolute type that are BK-spaces including the spaces c 0 and c and prove that the spaces e r 0 and e c r are linearly isomorphic to the spaces c 0 and c, respectively. Furthermore, some inclusion theorems are presented. Moreover, the α-, β-, γ-and continuous duals of the spaces e r 0 and e c r are computed and their bases are constructed. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes e c r p : ( ) and e c c r : ( ) are established, and characterizations of some other classes of infinite matrices are also derived by means of a given basic lemma, where 1 ≤ p ≤ ∞.

In this study, we define the double sequence spaces BS, BS(t), CS p , CS bp , CS r and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CS bp and the β(ϑ)-duals of the spaces CS bp and CS r of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CS bp , CS r and CS p of double series.  2004 Elsevier Inc. All rights reserved.

The matrix domain of the particular limitation methods Cesàro, Riesz, difference, summation and Euler were studied by several authors. In the present paper, certain topological properties and β-and γ -duals of the domain of a triangle matrix in a sequence space have been examined as an application of the characterization of the related matrix classes. Preliminaries, background and notationBy a sequence space, we understand a linear subspace of the space w = C N of all complex sequences which contains φ, the set of all finitely non-zero sequences, where N = {0, 1, 2, . . .}. We write ∞ , c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Also by p , we denote the space of all absolutely p-summable sequences, where 1 p < ∞. bv is the space consisting of all sequences (x k ) such that (x k − x k+1 ) in 1 and bv 0 is the intersection of the spaces bv and c 0 . We shall assume throughout unless stated otherwise that p, q > 1 with p −1 + q −1 = 1 and 0 < r < 1, and use the convention that any term with negative subscript is equal to naught.Let A = (a nk ) be an infinite matrix of complex numbers a nk , where n, k ∈ N, and write