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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 $\theta_h=f_h(\lambda)$..

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Depends on whether or not you want $U_\alpha$ to be open. –  JSchlather Jan 10 at 9:36
@JacobSchlather, yes, open is desirable. But any example would do. –  mpiktas Jan 10 at 9:41
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