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

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