TITLE: On the density of $C[0,1]$ in the space $L^{\infty}[0,1]$
QUESTION [11 upvotes]: It's easy to show $C[0,1]$ is not dense in $L^{\infty}[0,1]$ in the norm topology, but $C[0,1]$ is dense in $L^{\infty}[0,1]$ in the weak*-topology when take $L^{\infty}$ as the dual of $L^{1}$. how to prove it?

REPLY [12 votes]: Let $\psi_n$ be a sequence of standard mollifiers, symmetric about 0.  If $f \in L^\infty$, $g \in L^1$, it is a quick application of Fubini's theorem to see that $\int (f * \psi_n) g = \int f (\psi_n * g)$, where $*$ denotes convolution.  Since $f * \psi_n$ is continuous and $\psi_n * g \to g$ in $L^1$, we are done.