Let $\Omega \subseteq \mathbb{R}^n$ be open, $T \in \mathcal{D\,}'(\Omega)$ a distribution i.e. a linear and continous functional on test function space $\mathcal{D}(\Omega)$. I came across the following definition of $\nabla T$:
$$\langle\nabla T,\phi\rangle:=-\langle T,\nabla\phi\rangle \;\; \text{for every} \;\; \phi \in \mathcal{D}(\Omega).$$
This definition confuses me because $\nabla \phi$ is a vector valued function and $T$ should act on scalar valued functions as $\mathcal{D}(\Omega)$ elements.
The context is integration by parts. After the definition follow:
if $T=T_u$ with $u \in W^{1,p}(\Omega)$ then we have
$$\int_\Omega (\nabla u) \phi dx=-\int_\Omega u (\nabla \phi) dx \;\; \text{for every} \;\; \phi \in \mathcal{D}(\Omega)$$ which is a particular case of the definition.