How does the Hahn-Banach theorem implies the existence of weak solution? I came across the following question when I read chapter 17 of Hormander's book "Tha Analysis of Linear Partial Differential Operators", and the theorem is
Let $a_{jk}(x)$ be Lipschitz continuous in an open set $X\subset\mathbb{R}^n$, $a_{ij}=a_{ji}$, and assume that $(\Re a_{ij}(x))$ is positive definite.  Then
$$
\sum_{ij} D_j(a_{jk}D_ku)=f
$$
has a solution $u\in H_{(2)}^{loc}(X)$ for every $f\in L_{loc}^2(X)$
The auther then says if we can show that 
$$
|(f,\phi)|\leq \|M \cdot\sum_{ij}  D_j(\bar{a_{jk}}D_k\phi) \|_{L^2}, \quad \phi\in C_c^{\infty}(X)
$$
for some positive continuous function $M$, then by Hahn-Banach theorem there exists some $g\in L^2$
$$
  (f,\phi)=\left(g,M\cdot\sum_{ij}  D_j(\bar{a_{jk}}D_k\phi)\right)
$$
which inplies that the weak solution is $u=Mg$. what confuses me is how the Hahn-Banach theorem is used here to show the existence of $g$.
Thanks for your help

Define the functional:
$$k(M Lw)=\int_{X} fw$$
where $L$ is the differential opeartor. 
This is a bounded functional thanks to estimate you are assuming, also notice that you use the estimate to check that $k$ is well defined .
Then thanks to the Hahn-Banach theorem you can extendthe domain of this functional to the whole $L^{2}$ as nullUser mention.
Finally you use  Riesz-Representation theorem to obtain the solution.
