Norm closure of classical pseudodifferential operators does not contain H\"ormander's class

The L2-norm closure of the algebra of all zero-order classical (or polyhomogeneous) pseudodifferential operators on a closed manifold $X$ is proven not to contain H\"ormander's class $\Psi^0(X)$. A set of generators (for that closed algebra) smaller than all zero-order classical pseudos is also described.