Let $\psi: l^2 \to \mathbb{R}$, I can't understand what this stands for
$$<\psi, e_j>=\frac{(-1)^j}{j!}\in\mathbb{R}$$
$e_j$ are zero vectors with a $1$ in the $j$-th position.
What informations it gives me on $\psi$?
If $\psi$ was a sequence I would say that the product is the standard inner product of $l^2$ elements, so the result is the $j$-th element of the sequence $\psi$.
Otherwise $\psi$ is a function, what does it mean?
$<\psi,x>$ is just another way of writing $\psi(x)$.Therefore the above expression gives you the value of the function at all of the $e_j's$. Moreover, if $e_j's$ are the basis elements and $\psi$ is a linear function then it completely defines $\psi$ on the vector space.
Hope it is helpful