Dual of L infinity is larger than L1
= Dual of L infinity is larger than L1
{c}
{title2=$(L^\infty)^*\supsetneq L^1$}
On $\mathbb R^n$, integration embeds $L^1$ isometrically into $(L^\infty)^*$, but this is not surjective. For example, the functional taking the limit at infinity on $\mathbb C1\oplus C_0(\mathbb R^n)$ extends to $L^\infty$ by the <Hahn-Banach theorem>; it vanishes on compactly supported functions and is one on the constant function, so no $L^1$ density represents it.