TITLE: $L^1$ and $L^{\infty}$ are not reflexive
QUESTION [19 upvotes]: I want some proof for the following statement :
$L^1$ and $L^{\infty}$ are not reflexive.
Can anyone help me, please? or reference me?

REPLY [7 votes]: Brezis "Fuctional analysis Sobolev spaces and Partial Differential Equations" pag 99-102
There is everything you need