  We study the potential functions that determine the optimal density for
$\varepsilon$-entropically regularized optimal transport, the so-called
Schr\"odinger potentials, and their convergence to the counterparts in
classical optimal transport, the Kantorovich potentials. In the limit
$\varepsilon\to0$ of vanishing regularization, strong compactness holds in
$L^{1}$ and cluster points are Kantorovich potentials. In particular, the
Schr\"odinger potentials converge in $L^{1}$ to the Kantorovich potentials as
soon as the latter are unique. These results are proved for all continuous,
integrable cost functions on Polish spaces. In the language of Schr\"odinger
bridges, the limit corresponds to the small-noise regime.
