Chain rule in a Hilbert space.

Question

Let $F:H\to \mathbb{R}$ be some functional on a Hilbert space $H$. Denote its Frechet derivative at $h\in H$ as $\frac{\delta F}{\delta h}(h)$. Suppose $h_t$ is a curve in $H$ i.e $$h_\cdot : \mathbb{R}\to H,~~~~ t\mapsto h_t$$

how do I use the chain rule in this case to show that

$$ \frac{\partial}{\partial t}F(h_t)=\Big\langle \frac{\delta F}{\delta h}(h_t) , \partial_t h_t \Big\rangle ~~~~~?$$

Answer 1

Hint:

Using the composition $$\mathbb R\stackrel h\to H \stackrel F\to\mathbb R$$ and the point of view of $$\mathbb R\stackrel{F\cdot h}\longrightarrow\mathbb R$$ it means that $(F\circ h)'=F'(h)\cdot h'$ among the derivatives.