Is it possible to construct and atlas for space $\ell_1$? I.e. give an example of collection $(U_\alpha,\phi_\alpha)$, such that $\cup U_\alpha=\ell_1$ and $\phi_\alpha:U_\alpha\to R^{n}$ is a bijection and differentiable in some sense.
Note I mix a bit the definitions of the atlas for topological space and atlas for Banach manifolds. What I am actually interested is saying that if I have a following model
$$y_t=\sum_{h=0}^\infty \theta_hx_{t-h}+\varepsilon_t,$$
with $(\theta_h)\in \ell_1$, I can always find a collection of double differentiable functions $f_h(\lambda)$ (\lambda\in \mathbb{R}^k) such that $\theta_h=f_h(\lambda)$..