In this note we treat some open problems of HEINRICH on ultrapowers of
locally convex spaces. In section 1 we investigate the localization o
f bounded sets in the full ultrapower of a locally convex space, in pa
rticular the coincidence of the full and the bounded ultrapower, mainl
y concentrating in the case of (DF)-spaces. In section 2 we provide a
partial answer to a question of HEINRICH on commutativity of strict in
ductive limits and ultrapowers. In section 3 we analyze the relation b
etween some natural candidates for the notion of superreflexivity in t
he setting of Frechet spaces. We give an example of a Frechet-Schwartz
space which is not the projective limit of a sequence of superreflexi
ve Banach spaces.