Some new paranormed difference sequence spaces and weighted core

1. Preliminaries, background and notation By ω, we shall denote the space of all real valued sequences. Any vector subspace of ω is called as a sequence space. We shall write l∞, c and c0 for the spaces of all bounded, convergent and null sequences, respectively. Also by bs, cs, l1 and lp, we denote the spaces of all bounded, convergent, absolutely and p-absolutely convergent series, respectively, 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., |αn − α| → 0 and g(xn − x) → 0 imply g(αnxn − 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 (pk) be a bounded sequences of strictly positive real numbers with sup pk = H and M = max{1,H}. Then, the linear spaces c(p), c0(p), l∞(p) and l(p)were defined byMaddox [1,2] (see also [3,4]) as follows: c(p) =  x = (xk) ∈ ω : lim k→∞ |xk − l|pk = 0 for some l ∈ C  , c0(p) =  x = (xk) ∈ ω : lim k→∞ |xk|k = 0  , l∞(p) =  x = (xk) ∈ ω : sup k∈N |xk|k < ∞  and l(p) =  x = (xk) ∈ ω : 