Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\infty$ which is stable under this operation, then its weak* closure is too. In other words, the weak* closure of any vector lattice in $l^\infty$ is a vector lattice. The proof is by transfinite induction! Does anyone know an easier proof of this simple fact?

Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\infty$ which is stable under this operation, then its weak* closure is too. In other words, the weak* closure of any vector lattice in $l^\infty$ is a vector lattice. The proof is by transfinite induction! Does anyone know an easier proof of this simple fact?