Proving that dual space of $L^\infty(\mathbb{R})$ is not separable

Question

This is the second part of a problem in Stein and Shakarchi's Functional Analysis.

I proved in the first part that $L^\infty(\mathbb{R})$ is not separable by constructing for each $a \in \mathbb{R}$ an $f_a \in L^\infty$, with $\|f_a - f_b\| \geq 1$ if $a \neq b$.

I tried similar trick to dual space of $L^\infty(\mathbb{R})$, but I don't think I have enough intuition about the space to show that it is not separable. Can anyone help me out?

Thank you so much!

Answer 1

If the dual of $L^{\infty}$ were separable, then $L^{\infty}$ would be separable as well.

The fact that if $X^*$ is separable, also $X$ is follows from the Hahn-Banach theorem. Here is a proof of this fact.