SIMPLE BOUNDED WEIGHT MODULES OF $$ \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right),\kern0.5em \mathfrak{sp}\left(\infty \right) $$

We classify the simple bounded weight modules of the Lie algebras $$ \mathfrak{sl}\left(\infty \right),\kern0.5em \mathfrak{o}\left(\infty \right) $$ sl ∞ , o ∞ and $$ \mathfrak{sp}\left(\infty \right) $$ sp ∞ , and compute their annihilators in $$ U\left(\mathfrak{sl}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{o}\left(\infty \right)\right),\kern0.5em U\left(\mathfrak{sp}\left(\infty \right)\right) $$ U sl ∞ , U o ∞ , U sp ∞ , respectively.