About the dual space of $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its relations to $H^{-1}(\Omega)$.

Question

I read about some things about $V=\{u\in H_0^1(\Omega): \text{div}u=0 \}$ and its dual space and I began to mix some of these things together. As a result: irritation. I hope you can help me out. First off, $V$ has the same topology as $H_0^1$ hence it is a Hilbert space with the scalar product $(u,v)_V=\int_\Omega \nabla u \cdot \nabla v dx$.

1. Of course $V \subset H_0^1$ and $V$ is continuously embedded in $H_0^1$ as $\|v\|_{H_0^1}=\|v\|_V$ for all $v \in V$. Hence we also know that $H^{-1}$ is a subspace of $V'$.
2. Girault & Raviart use in their book [Finite Element Approximation of the Navier-Stokes Equations] in Theorem 1.2 on page 158 that $f \in L^2(0,T;V')$, but by assumption we only know $f \in L^2(0,T;H^{-1})$. Hence they seem to use $L^2(0,T;H^{-1})\subset L^2(0,T;V')$. But for this wouldn't I need that $H^{-1}$ is continuously embedded in $V'$? To have the existence of a constant $C>0$ such that $$\|u\|_{L^1(0,T;V')}=\int_0^T \|u\|_{V'} dt \leq \int_0^T C \|u\|_{H^{-1}} dt =\|u\|_{L^1(0,T;H^{-1})}.$$
3. J. Simon proves in his paper [On the existence of the Pressure for Solutions of the Variational Navier-Stokes Equations]: '$H^{-1}(\Omega)^d$ and $V'$ themselves cannot be imbedded in the same Hausdorff space'. (p.226, 4th line)

Now, I see a contradiction between [2.] and [3.]. If we have [2.] then $H^{-1}$ is continuously embedded in $V'$. Hence $H^{-1}$ and $V'$ can both be embedded in a Hausdorff-space, namely $V'$ (which is even a Banach space as it is the dual of a Hilbert space I thought).

EDIT: After the great help in the comments, I know that $H^{-1}$ is continuously embedded in $V'$. Hence, Girault and Raviart can of course use the fact $L^2(0,T;H^{-1})\subset L^2(0,T;V')$. Now I look again at the paper of J. Simon and try to find out what he means. I will edit it here if I find something. Otherwise, I am always thankful for comments, answers and hints.

EDIT2: Okay, I think I know the answer. The problem lies in the word "embedding" that I totally mixed, duh. Girault and Raviart only use that the restriction operator from $H^{-1}$ to $V'$ is continuous - nothing more! But Simon includes injectivity in his definition of imbedding. But, since $V$ is not dense in $H_0^1$, $H^{-1}$ can't be injective in $V'$ (with the restriction operator).

If there are no objections in the next few days, I will post EDIT2 as an answer.

Answer 1

Okay, I think I know the answer. The problem lies in the word "embedding" that I totally mixed.

Girault and Raviart only need that the operator from $H^{-1}$ to $V'$ is continuous - nothing more! But Simon includes injectivity in his definition of imbedding. But, since $V$ is not dense in $H_0^1$, $H^{-1}$ is not injective in $V'$ (with the restriction operator).