Reflexivity of Lp spaces

For $1<p<\infty$, the identifications $(L^p{)}^{\ast }=L^q$ and $(L^q{)}^{\ast }=L^p$ make $L^p$ reflexive. On a non-atomic space such as ${\mathbb{R}}^n$, neither $L^1$ nor $L^{\infty }$ is reflexive.