The subdifferential of a lower
semicontinuous proper convex function on a Banach space is a maximal monotone
operator, as well as a maximal cyclically monotone operator. This result was
announced by the author in a previous paper, but the argument given there
was incomplete; the result is proved here by a different method, which is
simpler in the case of reflexive Banach spaces. At the same time, a new
fact is established about the relationship between the subdifferential of a
convex function and the subdifferential of its conjugate in the nonreflexive
case.