I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not:
Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ and $f:H\to\mathbb R$ be Fréchet differentiable. By definition, the Fréchet derivative ${\rm D}f$ of $f$ is a mapping $H\to H'$$^1$. By Riesz' representation theorem, for each $L\in H'$ we can find a unique $v\in H$ with $$Lu=\langle u,v\rangle\;\;\;\text{for all }u\in H\;.$$ So, we should be able to identify ${\rm D}f$ with a mapping $H\to H$ such that for all $x\in H$ $${\rm D}f(x)u=\langle u,{\rm D}f(x)\rangle\;,\tag 1$$ where ${\rm D}f(x)$ is considered as an element of $H'$ and $H$ on the left-handed and right-handed side, respectively.
How would we state $(1)$ for the second derivative ${\rm D}^2f$?
$^1$ Let $H'$ denote the topological dual space of $H$.
This is what in finite dimension is called the gradient of $f$, if it exists (and which you may call gradient in this case as well). It's the same idea as in the finite dimensional case. In order for this to work you need a natural isomorphism between the vector space and it's dual (which you usually don't have but) which is induced by the scalar product in case of a Hilbert space.