# Confusion in notation of representation of Bastiani derivative

In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions

Definition II.2. Let $$U$$ be an open subset of a Hausdorff locally convex space $$E$$ and let $$f$$ be a nap from $$U$$ to a Hausdorff locally convex space $$F$$. Then $$f$$ is said to have a derivative at $$x \in U$$ in the lirection of $$v \in E$$ if the following limit exists: $${ }^{.3}$$ $$D f_x(v):=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t} .$$
Definition II.3. Let $$U$$ be an open subset of a Hausdorff locally convex space E and let fe a map from $$U$$ to a Hausdorff locally convex space $$F$$. Then $$f$$ is Bastiani differentiable on $$U$$ [denoted by $$\left.f \in C^1(U)\right]$$ iff has a Gâteaux differential at every $$x \in U$$ and the map $$D f: U \times E \rightarrow F$$ defined by $$D f(x, v)=D f_x(v)$$ is continuous on $$U \times E$$.

At page 24 they state the following

Lemma VI.2. Let $$U$$ be an open subset of $$C^{\infty}(M)$$ and $$F: U \rightarrow \mathbb{K}$$ be Bastiani smooth. For every $$\varphi$$ such that the distribution $$D F_{\varphi} \in \mathcal{E}^{\prime}(M)$$ has an empty wave front set, there exists a unique function $$\nabla F_{\varphi} \in \mathcal{D}(M)$$ such that $$D F_{\varphi}[h]=\int_M \nabla F_{\varphi}(x) h(x) d x$$

Here $$\mathcal{D}(M)$$ is the space of test function on $$M$$ and $$\mathcal{E}^{\prime}(M)$$ is the dual of the space of section $$\Gamma (E)$$

Since $$h(x)$$ is a section in Lemma VI.2, how can we produce a number by multiplying it by a function and then intergrate it ?

Am I missing something?