  In [2] we characterized in terms of a quadratic growth condition various
metric regularity properties of the subdifferential of a lower semicontinuous
convex function acting in a Hilbert space. Motivated by some recent results in
[16] where the authors extend to Banach spaces the characterization of the
strong regularity, we extend as well the characterizations for the metric
subregularity and the strong subregularity given in [2] to Banach spaces. We
also notice that at least one implication in these characterizations remains
valid for the limiting subdifferential without assuming convexity of the
function in Asplund spaces. Additionally, we show some direct implications of
the characterizations for the convergence of the proximal point algorithm, and
we provide some characterizations of the metric subregularity and calmness
properties of solution maps to parametric generalized equations.
