  We show how to construct the nonstandard hull of certain infinite-dimensional
Lie algebras in order to generalize a theorem of Pestov on the enlargeability
of Banach-Lie algebras. In the process, we consider a nonstandard smoothness
condition on functions between locally convex spaces to ensure that the induced
function between the nonstandard hulls is smooth. We also discuss some
conditions on a function between locally convex spaces which guarantee that its
nonstandard extension maps finite points to finite points.
