In this paper we define the sequence spaces S ℓ ∞ ( p ) , S c ( p ) and S c 0 ( p ) and determine the Köthe-Toeplitz duals of S ℓ ∞ ( p ) . We also obtain necessary and sufficient conditions for a matrix A to map S ℓ ∞ ( p ) to ℓ ∞ and investigate some related problems.