In this paper, we show that under the condition $ 1<p_-, q_-, p_+, q_+<\infty$, the space $\ell^{ q(\cdot)} (L^{ p(\cdot)})$ is reflexive. In this way we give an answer to open problem posed by H\"ast\"o in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces $\ell^{ q(\cdot)} (L^{ p(\cdot)})$. What is important here is that the dual space of $\ell^{ q(\cdot)} (L^{ p(\cdot)})$ is specified. As its direct corollary, we show that the corresponding Besov space $B^{ s(\cdot)}_{ p(\cdot)q(\cdot)}$ is reflexive.
Categories: math.FA