Konuralp Journal of Mathematics Volume 4 No. 2 pp. 132–148 (2016) c KJM ON THE PARANORMED TAYLOR SEQUENCE SPACES HACER BILGIN ELLIDOKUZO ˜ GLU AND SERKAN DEMIRIZ Abstract. In this paper, the sequence spaces t r 0 (p), t r c (p) and t r (p) of non- absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces t r 0 (p), t r c (p) and t r (p) are linearly isomorphic to spaces c 0 (p), c(p) and ℓ(p), respectively. Further- more, the α-,β- and γ-duals of the spaces t r 0 (p), t r c (p) and t r (p) have been computed and their bases have been constructed and some topological proper- ties of these spaces have been investigated. Besides this, the class of matrices (t r 0 (p): µ) has been characterized, where µ is one of the sequence spaces ℓ∞,c and c 0 and derives the other characterizations for the special cases of µ. 1. Introduction By w, we shall denote the space of all real-valued sequences. Any vector subspace of w is called a sequence space. We shall write ℓ ∞ ,c and c 0 for the spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, ℓ 1 and ℓ p , we denote the spaces of all bounded, convergent, absolutely and p−absolutely convergent series, respectively, where 1 <p< ∞. A linear topological space X over the real field R is said to be a paranormed space if there is a subadditive function g : X → R such that g(θ)=0,g(x)= g(−x) and scalar multiplication is continuous, i.e., |α n − α|→ 0 and g(x n − x) → 0 imply g(α n x n − αx) → 0 for all α’s in R and all x’s in X, where θ is the zero vector in the linear space X. Assume here and after that (p k ) be a bounded sequences of strictly positive real numbers with sup p k = H and L = max{1,H}. Then, the linear spaces ℓ ∞ (p),c(p),c 0 (p) and ℓ(p) were defined by Maddox [12] (see also Simons [14] and 2000 Mathematics Subject Classification. 46A45, 40C05, 46B20. Key words and phrases. Taylor sequence spaces, matrix domain, matrix transformations. 132