According to James's theorem, a Banach space X is a reflexive space if and only if $$ \forall f \in X^* ~ \exists x (\| x \| = 1): |f(x)| = \| f \|. $$ It is known that $L_1 [0; 1]$ is not reflexive. But I can't find $F(x) \in (L_1 [0; 1])^*$ such that it does not attain its norm on the closed unit ball $B_1 (0)$ in $L_1 [0;1]$, I will be glad to see such an example.
P.S. I know that any $F(x) \in (L_1 [0; 1])^* $ is represented as: $$ F(x) = \int_0^1 f(t) x(t) dt $$ when $f(x) \in L_{\infty} [0 ; 1]$ and $\| F \| = \| f \|_{\infty}$