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.
Categories: math.FA