  In this paper, we reconsider the well-known result of Pego-Weinstein
\cite{MR1289328} that soliton solutions to the Korteweg-deVries equation are
asymptotically stable in exponentially weighted spaces. In this work, we
recreate this result in the setting of modern well-posedness function spaces.
We obtain asymptotic stability in the exponentially weighted space via an
iteration argument. Our purpose here is to lay the groundwork to use the
$I$-method to obtain asymptotic stability below $H^1$, which will be done in a
second, forthcoming paper \cite{PR}. This will be possible because the
exponential approach rate obtained here will defeat the polynomial loss in
traditional applications of the $I$-method \cite{MR1995945}, \cite{MR1951312},
\cite{pigottorb}.
