Let $F$ be a Hilbert space and $\Theta$ a $d$-dimensional closed Riemannian manifold. Consider the twice Fréchet differentiable functions $R \colon F \to [0, \infty)$ and $\phi \colon \Theta \to F$ and for $m \in \mathbb{N}_{> 0}$ the discrete objective function $$ F_m \colon (\Omega^*)^m \to \mathbb R, \qquad \big((r_k, \theta_k)\big)_{k = 1}^{m} \mapsto R\left(\frac{1}{m}\sum_{k = 1}^{m} h(r_k) \phi(\theta_k)\right) + \frac{\lambda}{m} \sum_{k = 1}^{m} h(r_k), $$ where $\Omega^* := (0, \infty) \times \Theta$.
On page 5 of Chizat - Sparse Optimization on Measures with Over-parametrized Gradient Descent (available on arXiv), it is claimed that as "gradients are characterised by the relation $\text{d} F_m(x)(\delta x) = \langle \nabla F_m(x), \delta x \rangle$, we get" a certain expression for $\nabla_{r_i} F_m$ and $\nabla_{\theta_i} F_m$, where $\langle \cdot, \cdot \rangle$ is a certain metric on $(\Omega^*)^m$. This claim is backed up by Wikipedia if I am not mistaken.
I want to verify those expressions so I tried to compute $\text{d}F_m(x)(\delta x)$, where, presumably, $x \big((r_k, \theta_k)\big)_{k = 1}^{m} \in (\Omega^*)^m$ and $\partial x \in T_x (\Omega^*)^m$. (As $(\Omega^*)^m$ and $\mathbb R$ are Riemannian manifolds, I understand this derivative as the derivative of a function between differentiable manifolds). To this end, I write $F_m = R \circ \Phi_m + \lambda H_m$, where $\Phi_m(x) := \frac{1}{m}\sum_{k = 1}^{m} h(r_k) \phi(\theta_k)$ and $H_m(x) := \frac{1}{m} \sum_{k = 1}^{m} h(r_k)$. Because of the sum structure, the derivative of $\Phi_m$ is the uniformly weighted sum of the derivatives of $h(r_k) \phi(\theta_k)$, for $k \in \{ 1, \ldots, m \}$, which I want to compute using the product rule.
BUT: What kind of derivative is $$ \text{d} \phi(x)(\delta x) $$ supposed to be? It can't be a Fréchet derivative $\text{D} f(x)[h] \in Y$, where $f \colon X \to Y$ and $x, h \in X$ and $X, Y$ are normed spaces (because $(\Omega^*)^m$ is not a normed space) because $x \in (\Omega^*)^m$ and $\partial x \in T_x (\Omega^*)^m$ (where $T_x M$ denotes the tangent space of $M$ at $x$) and it cannot be the derivative of a map between manifolds, as $F$ might be infinite-dimensional and hence not a manifold in the standard sense. (I am aware of, but don't know anything about Hilbert manifolds.)