Simple bounded weight modules of $displaystyle{mathfrak{sl}(infty)}$, $displaystyle{mathfrak{o}(infty)}$, $displaystyle{mathfrak{sp}(infty)}$

We classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.