Differentiation and Chain Rule on the Hilbert Space $L^2$. (Reisz Representation).

Question

Let $F:L^2(\mathbb{R}^d)\to \mathbb{R}$ be a functional on the Hilbert space $L^2(\mathbb{R}^d)$ and $\rho:\mathbb{R}\to L^2(\mathbb{R}^d)$ a curve in the space $L^2(\mathbb{R}^d)$.

I want to calculate $\frac{d}{dt}F(\rho(t))$. Let $DF(\rho(t))$ be the Frechet derivative at $\rho(t)$.

1. What `chain rule' do I apply to get $\frac{d}{dt}F(\rho(t))=DF(\rho(t))\dot{\rho}(t)$? (since this is not the usual chain rule in Euclidean space).
2. Since $DF(\rho(t))$ is a linear functional on $L^2(\mathbb{R}^d)$ it can be associated to an element $f$ of $L^2(\mathbb{R}^d)$ so that $DF(\rho(t))\dot{\rho}(t)=\langle f , \dot{\rho}(t) \rangle_{L^2(\mathbb{R}^d)}$. Is $f$ known in this case, does it relate to this Wiki article?

(note I have denoted the derivative of $\rho$ with respect to $t$ as $\dot{\rho}(t)$)

Answer 1

1. You apply the chain rule for the Frechet Derivative (https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative#Properties)
2. The so-called functional derivative is actually the Gateaux derivative. Wikipedia also explains the relations between the Frechet and Gateaux derivatives (https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative#Relation_to_the_Gateaux_derivative). If the Frechet derivative $DF(x)$ (and thus $\langle f, \cdot\rangle_{L^2(\mathbb{R}^d)}$) exists, then it is also the Gateaux derivative, the other way around is not true.

Answer 2

$$\frac{dF(\rho(t))}{dt} = A$$ is such that: $$ \frac{F(\rho(t+h))-F(\rho(t))-Ah}{h} = \frac{F(\rho(t)+\rho'(t)h+r_h)-F(\rho(t))- Ah}{h} = \frac{DF_{\rho(t)}(\rho'(t)h+r_h)+g_h- Ah}{h} \rightarrow DF_{\rho(t)}(\rho'(t))-A = 0$$

For the above to work, we need $DF_{\rho(t)}$ to be of bounded norm so that $\frac{||DF_{\rho(t)}r_h||}{h} \rightarrow 0$ and $\rho'(t)$ also to be of bounded norm so that $\frac{||g_h||}{h} \rightarrow 0$.

$\rho'(t)$ is such that, $$\lim_{h \rightarrow 0} \frac{||\rho(t+h)-\rho(t)-\rho'(t)h||_{L^2}}{h} = 0.$$