Hey, I have difficulties with the following question;

Let $\displaystyle H$ be a Hilbert space with associated inner product $\displaystyle \langle x,y\rangle$, and let $\displaystyle E$ be a linear subspace of $\displaystyle H$. Define the orthogonal set of $\displaystyle E$ as $\displaystyle E^*=\{y\in H| \langle y,x\rangle=0 ~ ~\forall x\in E\}$. Show that $\displaystyle E$ is dense in $\displaystyle H$ if and only if $\displaystyle E^*=\{0\}$.

I've already shown that $\displaystyle E^*$ is always a closed linear subspace of $\displaystyle H$, and I am told I can use the fact that, if a given linear subspace $\displaystyle M$ is closed, then $\displaystyle (M^*)^*=M$.

From here, I guess that I have to sow that $\displaystyle E^*=\{0\}$ iff the closure of $\displaystyle E$ is $\displaystyle H$, but I can't manage to get started on either side of the iif.