  We formalize the observation that the same summability methods converge in a
Banach space $X$ and its dual $X^*$. At the same time we determine conditions
under which these methods converge in the weak and weak*-topologies on $X$ and
$X^*$ respectively. We also derive a general limitation theorem, which yields a
necessary condition for the convergence of a summability method in $X$. These
results are then illustrated by applications to a wide variety of function
spaces, including spaces of continuous functions, Lebesgue spaces, the disk
algebra, Hardy and Bergman spaces, the BMOA space, the Bloch space, and de
Branges-Rovnyak spaces. Our approach shows that all these applications flow
from just two abstract theorems.
