Positivity 12 (2008), 363–374
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020363-12, published online January 11, 2008
On Semicomplete Cones
Fatimetou mint El Mounir
Abstract. We give here characterizations of relatively bounded maps, of
normal cones, nuclear and well based cones. We also discuss two notions of
semi-complete considered by Mokobodzki.
AMS Subjects Classiﬁcation (2000). 46A40.
Keywords. Normal cones, nuclear cones, semicomplete cones, regular cones,
well based cones.
It is known that the normality of convex cones is a fundamental notion used in
the theory of ordered topological vector spaces. The notion of nuclear cone, intro-
duced by G. Isac () has interesting applications, 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 variational principles for vector-valued functions
(, , ).
In this paper, we consider a class of convex cones possessing some property
(P ). This class contains semicomplete cones in the sense of Mokobodzki, in which
zero has a countable basis of neighborhoods. First of all, we characterize relatively
bounded maps on these cones. We obtain in particular that each relatively bounded
map can be majorized by a continuous sminorm on the space and increasing on the
cone (Proposition 1.1). This is an amelioration of Proposition 3.2 of  were it is
showed that each relatively bounded map on a C-regular cone in which zero has a
countable basis of neighborhoods can be majorized by a continuous seminorm. If,
furthermore, it is linear, then it can be majorized by a continuous linear functional
on the space which is positive on the cone (Proposition 1.5).
These results allow on one hand to obtain many characterizations of nor-
mality and nuclearity of cones (Propositions 2.1, 2.2 and 3.1). A necessary and
suﬃcient condition for well based cones is also obtained. On the other hand, they
allow to compare these notions for distinct topologies on the space. In particular,
we show that if the cone K is normal (resp. nuclear, resp. well based) with respect