Corollary

— For every number p such that $1 < p < +\infty$, the Banach space $L^p(T, \mu)$ is reflexive.

In general, the dual of $L^\infty$ is not isomorphic to $L^1$, consequently $L^1$ and $L^\infty$ are not reflexive (Exer. 10). We are going to characterize the continuous linear forms on $L^\infty$ that arise, by passage to the quotient, from a linear form $g \mapsto \int f g d\mu$ on $\mathscr{L}^\infty$, where $g \in \mathscr{L}^1$.

The ordered vector space $L^\infty(T, \mu)$, which is a subspace of $L^1_{loc}(T, \mu)$, is fully lattice-ordered; for, if $(f_\alpha)$ is a family of positive functions in $\mathscr{L}^\infty$ whose set of classes $(\dot{f}_\alpha)$ is bounded above in $L^\infty$, there exists an $a \geq 0$ such that $N_\infty(f_\alpha) \leq a$ for all $\alpha$. Since $L^1_{loc}(T, \mu)$ is fully lattice-ordered, the family $(\dot{f}_\alpha)$ admits a supremum $\dot{h}$ in $L^1_{loc}(T, \mu)$; but since $\dot{a} \geq \dot{f}_\alpha$ for every $\alpha$, we have $\dot{h} \leq \dot{a}$, consequently $N_\infty(h) \leq a$, whence our assertion.