Let $\Psi: \mathcal{l}_1 \rightarrow \mathcal{l}_1$ such that $\Psi(u) = v$, where $\forall n \in \mathbb{N}$: $$v_n = \sum^{n}_{p=1} u_pu_{n-p}$$ 1) Show that $\Psi$ is well defined.
2) Show that $\Psi$ is $\mathcal{C}^1$ and calculate its differential.
So just answering the first question I struggled a lot. By using the summation by parts, I managed to proove that the series $\sum^{n}_{p=1} u_pu_{n-p}$ converges, yet as we are working in $l_1$ I have to additionally show that $\sum |v_n|$ converges, which I am unable to do.
As for the second, I managed to notice that if we take $a,h \in l_1$ we then have $$ \Psi (a+h) = (\sum^{n}_{k=1}a_ka_{n-k})_{n \in \mathbb{N}} + (\sum^{n}_{k=1}h_kh_{n-k})_{n \in \mathbb{N}} + 2((\sum^{n}_{k=1}a_kh_{n-k}+h_ka_{n-k})_{n \in \mathbb{N}}) $$ Thus the differential pf this function would be equal to $$2((\sum^{n}_{k=1}a_kh_{n-k}+h_ka_{n-k})_{n \in \mathbb{N}})$$ Which is linear, and then I have to show that $\Psi$ is differentiable, but whatever I do doesn't make much sens to me.