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.