  We introduce the notion of an asymptotically equicontinuous sequence of
linear operators, and use it to prove the following result. If $X,Y$ are
topological vector spaces, if $T_n,T:X\to Y$ are continuous linear maps, and if
$D$ is a dense subset of $X$, then the following statements are equivalent: (i)
$T_nx\to Tx$ for all $x\in X$, and (ii) $T_n x\to Tx$ for all $x\in D$ and the
sequence $(T_n)$ is asymptotically equicontinuous.
