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

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