Summary:

*I. J. Maddox* defined the sequence spaces

${\ell}_{\infty}\left(p\right),c\left(p\right)$ and

${c}_{0}\left(p\right)$ in [Proc. Camb. Philos. Soc. 64, 335–340 (1968;

Zbl 0157.43503), Q. J. Math., Oxf. (2) 18, 345–355 (1967;

Zbl 0156.06602)]. In the present paper, the sequence spaces

${a}_{0}^{r}(u,p)$ and

${a}_{c}^{r}(u,p)$ of non-absolute type are introduced and it is proved that the spaces

${a}_{0}^{r}(u,p)$ and

${a}_{c}^{r}(u,p)$ are linearly isomorphic to the spaces

${c}_{0}\left(p\right)$ and

$c\left(p\right)$, respectively. Besides this, the

$\alpha $-,

$\beta $- and

$\gamma $-duals of the spaces

${a}_{0}^{r}(u,p)$ and

${a}_{c}^{r}(u,p)$ are computed and their bases are constructed. Finally, a basic theorem is given and some matrix mappings from

${a}_{0}^{r}(u,p)$ to the sequence spaces of Maddox and to new sequence spaces are characterized.