For each $p \in \mathbb{R}$, we consider $l_p (\mathbb{Z}) := \{ f : \mathbb{Z} \rightarrow \mathbb{R} : \sum_{i = 0}^{\infty} |f(i)^p| < \infty \}$. I am trying to figure out how a certain conglomeration (a restricted product $A = \Pi'_p l_p(\mathbb{Z})$ of these spaces could work, one which features continuous linear maps $g_p : l_p(\mathbb{Z}) \rightarrow A$.
This could be to do with the Berkovitch completion. Potentially it could be an Ind-limit of tensor products $\otimes_{p_1, ...,p_n} l_{p_i}$, but this seems unlikely to work.
I am hoping that $A$ projects onto each $l_p(\mathbb{Z})$ as well.
Is this possible, and if not, what are some insights into how it is not possible to make a conglomeration such as this?