Thai Journal of Mathematics Volume 8 (2010) Number 2 : 311–329 www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 On the Spaces of λ-Convergent and Bounded Sequences M. Mursaleen 1 and A.K. Noman Abstract : In the present paper, we introduce the notion of λ-convergent and bounded sequences. Further, we define some related BK spaces and construct their bases. Moreover, we establish some inclusion relations concerning with those spaces and determine their α-, β- and γ -duals. Finally, we characterize some related matrix classes. Keywords : Sequence spaces; BK spaces; Schauder basis; α-, β- and γ -duals; Matrix mappings. 2000 Mathematics Subject Classification : 40C05, 40H05, 46A45. 1 Introduction By w, we denote the space of all complex sequences. If x ∈ w, then we simply write x =(x k ) instead of x =(x k ) ∞ k=0 . Also, we shall use the conventions that e = (1, 1,...) and e (n) is the sequence whose only non-zero term is 1 in the n th place for each n ∈ N, where N = {0, 1, 2,...}. Any vector subspace of w is called a sequence space. We shall write ℓ ∞ , c and c 0 for the sequence spaces of all bounded, convergent and null sequences, respectively. Further, by ℓ p (1 ≤ p< ∞), we denote the sequence space of all p-absolutely convergent series, that is ℓ p = {x =(x k ) ∈ w : ∑ ∞ k=0 |x k | p < ∞} for 1 ≤ p< ∞. Moreover, we write bs, cs and cs 0 for the sequence spaces of all bounded, convergent and null series, respectively. A sequence space X is called an FK space if it is a complete linear metric space with continuous coordinates p n : X → C (n ∈ N), where C denotes the complex field and p n (x)= x n for all x =(x k ) ∈ X and every n ∈ N. A normed FK space is called a BK space, that is, a BK space is a Banach sequence space with continuous coordinates. 1 Corresponding author Copyright c  2010 by the Mathematical Association of Thailand. All rights reserved.