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.