Show that : $B_1(0,L^\infty((\Omega,\mathcal{F},\mathbb{P})))$ is compact in the weak topology $\sigma(L^1,L^\infty)$.

Question

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $B_1(0,L^\infty((\Omega,\mathcal{F},\mathbb{P})))$, the closed unit ball of $L^\infty((\Omega,\mathcal{F},\mathbb{P}))$.

Show that : $B_1(0,L^\infty((\Omega,\mathcal{F},\mathbb{P})))$ is compact in the weak topology $\sigma(L^1,L^\infty)$.

Hint :

We say that : $B_1(0,L^\infty((\Omega,\mathcal{F},\mathbb{P})))$ is compact in the weak topology $\sigma(L^\infty,L^1)$. Show that :

$$ \begin{array}{lll} i: & (L^\infty ,\sigma(L^\infty,L^1))&\longrightarrow& (L^1 ,\sigma(L^1,L^\infty)) \\ &f &\longrightarrow & f \end{array}$$ Is a continuous injection

Answer 1

This is an immediate consequence of Banach Alaoglu Theroem and the fact that $L^{\infty}$ is the dual of $L^{1}$.