We introduce the sequence space $l_p^{\lambda}$ of non-absolute type and prove that the spaces $l_p^{\lambda}$ and $l_p$ are linearly isomorphic for $0<p\leq\infty$. Further, we show that $l_p^{\lambda}$ is a $p$-normed space and a $BK$-space in the cases of $0<p<1$ and $1\leq p\leq \infty$, respectively. Furthermore, we derive some inclusion relations concerning the space $l_p^{\lambda}$. Finally, we construct the basis for the space $l_p^{\lambda}$, where $1\leq p\leq\infty$.