Of which function space is $L^\infty(\mathbb{R}^n)$ the dual space of

Question

In my notes there is a statement which says: By the Banach-Alaoglu theorem, the family $\{u_k\}$ is bounded in $L^\infty(\mathbb{R}^n)$, hence it is weak-star convergent to some $u\in L^\infty(\mathbb{R}^n)$. For bounded domains $\Omega$, $L^\infty(\Omega)$ is the dual of $L^1(\Omega)$, but $\mathbb{R}^n$ is not bounded, so the space of test functions is not $L^1(\mathbb{R}^n)$. What should it be then? (This space of test functions should at the very least contain $C^\infty_c(\mathbb{R}^n)$, because the context is looking for weak solutions to a certain PDE.) My worry is that $L^\infty(\mathbb{R}^n)$ is not the dual space of any function space; would the Banach-Alaoglu theorem still hold?

Answer 1

Who told you that $L^\infty(\Bbb R^n)$ is not the dual of $L^1(\Bbb R^n)$? In fact the duual of $L^1$ is $L^\infty$ in any $\sigma$-finite measure space.