Computable Hilbert space

Question

If $(H_{i},\|.\|,(e_{ij})_{j})$ be the computable Hilbert spaces for each $i \in \mathbb{N}$, what condition can be imposed on $H_{i}$'s so that the map $(i,j)\mapsto e_{ij}$ is $((\delta_{\mathbb{N}},\delta_{\mathbb{N}}),\delta_{H_{i}})$ computable? What i know is that since $(H_{i},\|.\|,(e_{ij})_{j})$ is computable Hilbert space for each $i \in \mathbb{N}$, the map $ j\mapsto e_{ij}$ is computable for each $i \in \mathbb{N}$. But computability on $i$ is also needed. Does the statement "computable sequence of computable Hilbert space $(H_{i})$" make sense?