Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality

We study the discrete Bourgain-Morrey sequence spaces $\ell^{p}_{q,r}(\mathbb{Z})$, recently introduced as discrete counterparts of Morrey-type spaces. We show that $c_{00}$ is dense in $\ell^{p}_{q,r}$, hence the spaces are separable. We establish embeddings $\ell^{1}\hookrightarrow \ell^{p}_{q,r}\hookrightarrow \ell^{r}$ for $r>1$, while for $r=1$ one has $\ell^{p}_{q,1}=\ell^{1}$. For each $p$, the identity $\ell^{p}_{q,p}=\ell^{p}$ yields uncountably many equivalent norms on $\ell^{p}$. We also introduce a block space as a natural predual of $\ell^{p}_{q,r}$ and prove the duality $(\ell^{p}_{q,r})^{*}=\mathrm{h}^{p'}_{q',r'}$, from which reflexivity follows for $1<p<q<\infty$ and $1<r<\infty$. This work completes the foundational stage of the discrete Bourgain-Morrey theory by fully characterizing its structure and duality.