  The Cauchy problem for the nonlinear Schr\"odinger equation is called
unconditionally well posed in a data space $E$ if it is well posed in the usual
sense and the solution is unique in the space $C([0,T]; E)$. In this paper,
this notion of unconditional well-posedness is redefined so that it covers
$L^p$-based Sobolev spaces as data space $E$ and it is equivalent to the usual
one when $E$ is an $L^2$-based Sobolev space $H^s$. Next, based on this
definition, it is shown that the Cauchy problem for the 1D cubic NLS is
unconditionally well posed in Bessel potential spaces $H^s_p$ for $4/3<p\le 2$
under certain regularity assumptions on $s$.
