# Thread: Proving sets are weakly compact

1. ## Proving sets are weakly compact

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

2. Originally Posted by calme
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 $L^\infty$ is weak*-compact (that is, compact in the $\sigma(L^\infty,L^1)$-topology). So all you have to do to prove that $B_1$ is weak*-compact is to show that it is weak*-closed in the unit ball of $L^\infty$. The result for general $B_n$ then follows by taking scalar multiples.

3. thank you a lot for your reply can you give me more details to prove that it's weakly closed can I use krein-smulitain theorem
it's really kind of you