Some Remarks on the Vector-Valued Variable Exponent Lebesgue Spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$

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 and uniformly convex. We also prove that when $1\le q_-,p_-,q_+,p_+<\infty,$ the convergence in norm implies the convergence in measure, and under some conditions on exponents, the approximation identity holds in the space $\ell^1(L^{\frac{p(\cdot)}{q(\cdot)}})$.