Hi,I want to prove that the sets
B_{n} ={g : 0 ≤ g ≤n} are weakly compact in σ(L^{∞},L^{1}).
Thank you for your help
The unit ball of $\displaystyle L^\infty$ is weak*-compact (that is, compact in the $\displaystyle \sigma(L^\infty,L^1)$-topology). So all you have to do to prove that $\displaystyle B_1$ is weak*-compact is to show that it is weak*-closed in the unit ball of $\displaystyle L^\infty$. The result for general $\displaystyle B_n$ then follows by taking scalar multiples.