Recover a function from the differential map

Question

Let $H$ be a Hilbert space, and suppose we have $f:H\longrightarrow B(H,\mathbb{R})$, where $B(H,\mathbb{R})$ denotes the set of linear and bounded functions from $H$ to $\mathbb{R}$. Then, given $u\in H$, we have $f(u)\in B(H,\mathbb{R})$. My question is:

Is it possible to obtain $F:H\longrightarrow\mathbb{R}$ continuous and Fréchet differentiable at every point of $H$ such that $DF:u\in H\longrightarrow DF(u)\in B(H,\mathbb{R})$ is equal to $f$, that is, $DF(u)=f(u)\ \forall\,u\in H$?

Answer 1

The question is if f has a potential. This is not true in general (even in finite dimentions). For example, if $H=\mathbb{R}^3$ then such an $F$ exists iff $\nabla\times f=0$.