This paper concerns a collection of sequence spaces we shall refer to as d α {d_\alpha } spaces. Suppose α = ( α 1 , α 2 , … ) \alpha = ({\alpha _1},{\alpha _2}, \ldots ) is a bounded number sequence and α i ≠ 0 {\alpha _i} \ne 0 for some i i . Suppose P \mathcal {P} is the collection of permutations on the positive integers. Then d α {d_\alpha } denotes the set to which the number sequence x = ( x 1 , x 2 , … ) x = ({x_1},{x_2}, \ldots ) belongs if and only if there exists a number k > 0 k > 0 such that \[ h α ( x ) = lub p ∈ P ⁡ ∑ i = 1 ∞ | x F ( i ) α i | > k . h_\alpha (x) = \operatorname {lub}_{p \in \mathcal {P}} \sum \limits _{i = 1}^\infty |x_{F(i)} \alpha _i| > k. \] h α h_\alpha is a norm on d α d_\alpha and ( d α , h α ) (d_\alpha , h_\alpha ) is complete. We classify the d α {d_\alpha } spaces and compare them with l 1 {l_1} and m m . Some of the d α {d_\alpha } spaces are shown to have a semishrinking basis that is not shrinking. Further investigation of the bases in these spaces yields theorems concerning the conjugate space properties of d α {d_\alpha } . We characterize the sequences β \beta such that, given α , d β , = d α \alpha ,{d_\beta }, = {d_\alpha } . A class of manifolds in the first conjugate space of d α {d_\alpha } is examined. We establish some properties of the collection of points in the first conjugate space of a normed linear space S S that attain their maximum on the unit ball in S S . The effect of renorming c 0 {c_0} and l 1 {l_1} with h α {h_\alpha } and related norms is studied in terms of the change induced on this collection of functionals.