  Using completions of certain universal enveloping algebras, we provide a
natural setting for families of defining relations for the principal subspaces
of standard modules for untwisted affine Lie algebras. We also use the theory
of vertex operator algebras and intertwining operators to construct exact
sequences among principal subspaces of certain standard
$\widehat{\mathfrak{sl}(n)}$-modules, $n \ge 3$. As a consequence, we obtain
the multigraded dimensions of the principal subspaces $W(k_1\Lambda_1 + k_2
\Lambda_2)$ and $W(k_{n-2}\Lambda_{n-2} + k_{n-1} \Lambda_{n-1})$. This
generalizes earlier work by Calinescu on principal subspaces of standard
$\widehat{\mathfrak{sl}(3)}$-modules.
