# Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out.

Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ where $e_n(t) = e^{i 2 \pi nt}$. Let $D$ in $\mathcal{L}(\operatorname{span}\{ e_n \}_{n=-\infty}^\infty)$ denote the operator of differentiation. Let $\mathcal{D} = \{ f \in L_2[0,1] : \sum_{n=-\infty}^\infty |n^2 (f,e_n)|^2 < \infty \}$. Show that there is a unique extension of $D^2$ from $\mathcal{D}$ to $L_2[0,1]$. Moreover, $D^2 : \mathcal{D} \to L_2[0,1]$ has closed graph in $L_2[0,1] \oplus L_2[0,1]$.

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Given $f=\sum_{n=-\infty}^{+\infty}(f,e_n)e_n\in\mathcal D$, do you see how to define $D^2$? –  Davide Giraudo Dec 16 '12 at 9:39