Positivity 11 (2007), 485–495
2007 Birkh¨auser Verlag Basel/Switzerland
Normality and Nuclearity of Convex Cones*
Fatimetou mint El Mounir
Abstract. We give here some characterizations for normality and nuclearity
of convex cones. We obtain a suﬃcient condition for weakly normal cone to
be normal (respectively nuclear).
Mathematics Subject Classiﬁcation (2000). 46A40.
Keywords. Normal cones, nuclear cones, well based cones.
The notion of nuclear cone, introduced by G. Isac () has interesting applica-
tions, not only to the study of Pareto eﬃciency in the theory of optimization of
vector-valued functions, or to the study of some problems in Functional Analysis
but, also it is a mathematical tool used now to obtain new Ekeland type varia-
tional principles for vector-valued functions (, , ). We present in this paper
simple characterizations (which are mostly ameliorations of classical conditions)
for a convex cone in which 0 has a countable basis of neighborhoods (respectively a
countable basis of weak neighborhoods) to be normal (respectively nuclear). In ,
McArthur has shown that in Fr´echet spaces, every weakly normal cone is normal.
We extend this property to all cones in which 0 has a countable basis of neighbor-
hoods. Similarly, we obtain the extension to all cones in which 0 has a countable
basis of weak neighborhoods of a result shown in  for cones which moreover are
weakly C-regular. Finally, we give some characterizations of nuclear cones in the
In this paper, (E,τ) will design a Hausdorﬀ topological vector space (t.v.s.).
If (E,τ) is a Hausdorﬀ locally convex space (l.c.s.), we denote by σ the weak
topology associated to the duality (E, E
). The set of neighborhoods of zero for
the topology τ (respectively σ) is denoted by ϑ
(0) (respectively by ϑ
* Work partially supported by FNARS, Project 003.