Báo cáo hóa học: Research Article On Some Generalized B m-Difference Riesz Sequence Spaces and Uniform Opial Property

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Báo cáo hóa học: " Research Article On Some Generalized B m-Difference Riesz Sequence Spaces and Uniform Opial Property"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 485730, 17 pagesdoi:10.1155/2011/485730Research ArticleOn Some Generalized B m-Difference RieszSequence Spaces and Uniform Opial Property ¨¨ Metin Basarır and Mahpeyker Ozturk ¸ Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey Correspondence should be addressed to Metin Basarır, basarir@sakarya.edu.tr ¸ Received 29 November 2010; Accepted 18 January 2011 Academic Editor: Radu Precup Copyright q 2011 M. Basarır and M. Ozturk. This is an open access article distributed under ¨¨ ¸ the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. q q q We define the new generalized difference Riesz sequence spaces r∞ p, B m , rc p, B m , and r0 p, B m q which consist of all the sequences whose B m -transforms are in the Riesz sequence spaces r∞ p , q q rc p , and r0 p , respectively, introduced by Altay and Basar 2006 . We examine some topological ¸ q q q properties and compute the α-, β-, and γ -duals of the spaces r∞ p, B m , rc p, B m , and r0 p, B m . Finally, we determine the necessary and sufficient conditions on the matrix transformation q q q from the spaces r∞ p, B m , rc p, B m , and r0 p, B m to the spaces l∞ and c and prove that q q have the uniform Opial property for pk ≥ 1 for all m m sequence spaces r0 p, B and rc p, B k ∈ Æ.1. IntroductionLet w be the space of real sequences. We write l∞ , c, c0 for the sequence spaces ofall bounded, convergent and null sequences, respectively. Also, by bs, cs, and l1 , wedenote the sequence spaces of all bounded, convergent and absolutely convergent series,respectively. A linear topological space X over the real field R is said to be a paranormed spaceif there is a subadditive function g : X → R such that g θ g −x and scalar 0, g xmultiplication is continuous; that is, |αn − α| → 0 and g xn − x → 0 imply g αn xn − αx → 0for all α’s in R and all x’s in X , where θ is the zero vector in the linear space X . Assumehere and after that p pk is a bounded sequence of strictly positive real numbers with max{1, H }. Then, the linear space l∞ p , c p , c0 p , and l p weresup pk H and Mdefined by Maddox 1, 2 , Nakano 3 and Simons 4 as follows:2 Journal of Inequalities and Applications xk ∈ w : sup|xk |pk < ∞ , l∞ p x k ∈Æ 0 for some l ∈ Ê , 1.1 xk ∈ w : lim |xk − l|pk cp x k→∞ xk ∈ w : lim |xk |pk c0 p x 0, k→∞which are the complete spaces paranormed by sup|xk |pk /M , g1 x k ∈Æ 1.2 pk xk ∈ w : n .Journal of Inequalities and Applications 3The Riesz sequence spaces introduced by Altay and Basar in 5, 6 are ¸ ⎧ ⎫ pk ⎨ ⎬ k 1 xk ∈ w : n .The results related to the matrix domain of the matrix B are more general and morecomprehensive than the corresponding consequences of matrix domain of Δ and includethem 6, 8–13 . Basarır and Kayikci 14 defined the matrix Bm bnk which reduced the difference m ¸ ¸ m−1matrix Δ ΔΔ in case r 1, s −1 by m ⎧⎛ ⎞ ⎪ ⎪⎝ m ⎠ m−n k n−k ⎨ ...