Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{ e_n \}$ and $f : H \to H$ be a sufficiently "nice" mapping so that it has the Frechet derivative $Df \in L(H,H)$.
Here, $L(H,H)$ is the space of bounded linear operators on $H$.
Now, given any subset $S \subset \{ e_n \}$, let $\pi_S : H \to \overline{\langle S \rangle }$ be the projection onto the closed subspace spanned by $S$.
Then, I wonder if $ D[\pi_S \circ f]= \pi_S\circ [Df]$ holds. Clearly, both $ D[\pi_S \circ f]$ and $\pi_S \circ [Df]$ belong to $L(H, \overline{\langle S \rangle})$ but I cannot figure out myself further..
Could anyone help me?