  In this paper, we investigate the geometric properties of the variable mixed
Lebesgue-sequence space $\ell^{q(\cdot)} (L^{p(\cdot)})$ as a Banach space. We
show that, if $ 1<q_-,p_-,q_+,p_+<\infty $, then $\ell^{q(\cdot)}
(L^{p(\cdot)})$ is strictly convex. We also prove that when $ 1\le
q_-,p_-,q_+,p_+<\infty, $ the norm convergence implies measure convergence and
the approximation identity holds in the space $ \ell^1(L^{\p\slash\q}) $.
