Dual space of a closed subspace of a Hilbert space

Question

I'm reading Girault and Raviart's book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument:

As $V=\{v\in H_0^1(\Omega)^N; {\rm div}\;v=0\; \text{in}\;\Omega\}$ is closed in $H_0^1(\Omega)^N$, then V* (dual space of V) can be identified with a subspace of $H^{-1}(\Omega)^N$, where $\Omega$ is an open set of $\mathbb{R}^N$.

My question is: why do we need that $V$ must be closed in $H_0^1(\Omega)^N$ (which it is) to get the isomorphism between V* and a subspace of $H^{-1}(\Omega)^N$? In fact, by using Hahn-Banach theorem we can build such isomorphism. (Remember that in order to use Hahn-Banach theorem, we just need a subspace of a vector space)

Thank you for your help!

Answer 1

For a counterexample to your claim, notice that $C^\infty_c(\Omega)$ is a subspace of $H^1_0(\Omega)$ (although it is not closed), and its dual is $D'(\Omega)$ (the space of all distributions on $\Omega$) which is not a subset of $H^{-1}(\Omega)$.

Note that to use the Hahn-Banach theorem, you need a sublinear function which is defined on the whole space and which dominates the functional you wish to extend on its domain. Apparently no such function exists for some distributions (for instance, the Dirac delta in dimension higher than 2).

Answer 2

For the identification (or related) see Theorem 4.9 in Rudin1991 ("Functional Analysis")

However, this identification (and also the closedness of $V$) need not be invoked for the particular result. I assume you are talking about Corollary 2.5 in Girault and Raviart. Namely, the Hahn-Banach theorem provides that any functional $\mathbf{f}\in V'$ that vanishes on $\mathscr{V}$ can be extended to a functional $\hat{\mathbf{f}}\in H^{-1}(\Omega)^N$ that vanishes on $\mathscr{V}$. Therefore, $\hat{\mathbf{f}}$ vanishes on $V$ and hence $\mathbf{f}$ vanishes on $V$ since $\hat{\mathbf{f}} = \mathbf{f}$ on $V$. This is the desired result. Now, the closedness of $V$ (i.e. $V$ being a Banach space) can be only needed if you want to invoke exactly the result (2.14) in Girault and Raviart, which is stated for Banach spaces. However, (2.14) holds also for general normed spaces.