[I was working on Stokes pde, and I'm stuck in proving this, couldn't find it anywhere]
Let $\Omega \in \mathbb{R}^N$ an open bounded connected set such as $\partial \Omega$ is $\mathcal{C}^1$.
And we define $$V=\{\vec v \in (H^1_0(\Omega))^N; \text{div} \vec v=0\}$$ With $\text{div} \vec v=\sum_{n=1}^N\frac{\partial v_i}{\partial x_i}$.
I'm stuck in proving : if $L$ is a linear continious form on $(H_0^1(\Omega))^N$ such as $L_{|V}=0$ then there exists a function $\phi \in L^2(\Omega)$, such as $$\forall \vec v\in (H_0^1(\Omega))^N : L(\vec v)=\int_{\Omega}\phi \;\text{div} \vec v \;dx$$
My idea : I proved that $V$ is a closed space of $(H_0^1(\Omega))^N$ then its a Hilbert space, and I tried to apply Riesz 's theorem, but ut doesn't work.