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Báo cáo hóa học: " Research Article Some Geometric Properties of Sequence Spaces Involving Lacunary Sequence"

Tham khảo luận văn - đề án 'báo cáo hóa học: " research article some geometric properties of sequence spaces involving lacunary sequence"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 81028 8 pages doi 10.1155 2007 81028 Research Article Some Geometric Properties of Sequence Spaces Involving Lacunary Sequence Vatan Karakaya Received 27 August 2007 Accepted 30 October 2007 Recommended by Narendra Kumar K. Govil We introduce new sequence space involving lacunary sequence connected with Cesaro sequence space and examine some geometric properties of this space equipped with Luxemburg norm. Copyright 2007 Vatan Karakaya. 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. 1. Introduction Let X II II be a real Banach space and let B X resp. S X be the closed unit ball resp. the unit sphere of X. A point x e S X is an H-point of B X if for any sequence xn in X such that xn 1 as n TO weak convergence of xn to x write xn x implies that ỊỊxn - x 0 as n TO. If every point in S X is an H-point of B X then X is said to have the property H . A point x e S X is an extreme point of B X if for any y z e S X the equality x y z 2 implies y z. A point x e S X is a locally uniformly rotund point of B X LUR-point if for any sequence xn in B X such that xn xỊỊ 2 as n TO there holds ỊỊxn - x 11 0 as n TO.A Banach space X is said to be rotund R if every point of S X is an extreme point of B X . If every point of S X is an LUR-point of B X then X is said to be locally uniformly rotund LUR . IfX is LUR then it has R-property. For these geometric notions and their role in mathematics we refer to the monograph 1-10 . By a lacunary sequence 0 kr where k0 0 we will mean an increasing sequence of nonnegative integers with kr - kr-1 TO as r- TO. The intervals determined by 0 will be denoted by Ir kr-1 kr . We write hr kr - kr-1. The ratio kr kr-1 will be denoted by qr. The space of lacunary strongly convergent sequences No was .