${\newcommand{\R}{\mathbb{R}}}$ Let $\ell^2 $ be the Hilbert space of square summable sequences of real numbers.
Consider a map $f: \R \to \ell^2$, that has components $f_n:\R\to \R$, i.e. $f(t)=\{f_n(t)\}_{n\in \mathbb N}$ so that $\sum_n(f_n(t))^2<+\infty$.
Motivation
If the derivative $\frac d {dt}f(t)$ exists, then it is given by $\{\partial_tf_n(t)\}_n\in \ell^2$ componentwise. However the existence of the derivatives $\{\partial_tf_n(t)\}_n\in \ell^2$ does not imply the existence of $\frac d {dt}f(t)$, because we need the remainder $$ R_n(t,h) = f_n(t+h) - f_n(t) - \partial_t f_n(t)h $$ to be $o(|h|)$ for $h\to 0$ uniformely in $n$.
I would like to know if adding a Sobolev-type assumption we obtain existence of the derivative, precisely:
Question:
Let $f$ as above, and suppose that the componentwise derivativee $\{\partial^{(k)}_t f_n\}_n \in \ell^2$ and the $\partial^{(k)}_t f_n$ are continuous for $k\leq 2$. Moreover assume that $$\int_\R \sum_n n^2|\partial_t^{(k)}f_n(t)|^2 dt <\infty$$ Does this imply that the derivative $\frac {d}{dt}f(t) \in \ell^2$ exists for all $t$? (and thus coincide with $\{\partial_t f_n\}_n$).