I have this exerciose:

Let $\Omega$ be a normed space. Prove that $\Omega$ is seperable if $\Omega^*$ is.

It is in the chapter with the Hahn-Banach theorem, so I think I should use that theorem.

**My attempt:**

We know that there is a countable set $K \subset \Omega^*$, such that for any $x^* \in \Omega^*$ there is a sequence of bounded lienar functionals in K, such that they converge to $x^*$ in the operator-norm.

So I need to associate the set $K\in \Omega^*$ with some set in $\Omega$, and show that this set is dense in $\Omega$.

The only thing I can think of is to start the other way around. For any $x \in \Omega$, we have the space $\{ax| a \in \mathbb{C}\}$, this is a subspace of $\Omega$, and it has a natural bounded linear functional $P_x(ax)=a$. by the Hahn-Banach theorem, this functional can be extended to a functional on the entire set $\Omega$, so we can assume that $P_x \in \Omega^*$.

Now we have a sequence in $\Omega^*$: $k_n\rightarrow P_x$, where $k_n \in K$, and the convergence is in the operator norm.

The problem now is to associate a proper element in $\Omega$ to each $k_n$, and show that this sequence converges to $x$.

Could this approach work, if so, how do I finish it? If not, could you please help me some other way?