Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let

• $E$ be a normed space
• $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space
• $f:E\to H$ be Fréchet differentiable
• $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,e_n\rangle\;\;\;\text{for }n\in\mathbb N$$

How can we compute the second Fréchet derivative ${\rm D^2}f_n:E\to\mathfrak L(E,\mathfrak L(E,H))$ of $f_n$?$^1$