Isac’s Cones

This is a very short research work representing an homage to the regretted Professor George Isac, Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. 17000, Kingston, Ontario, Canada, K7K 7B4. Professor Isac introduced the notion of “nuclear cone” in 1981, published in 1983 and called later as “supernormal cone” since it appears stronger than the usual concept of “normal cone”. For the first time, we named these convex cones as “Isac’s Cones” in 2009 , after the acceptance on professor Isac’s part. This study is devoted to Isac’s cones, including significant examples, comments and several pertinent references, with the remark that this notion has its real place in Hausdorff locally convex spaces not in the normed linear spaces, having strong implications and applications in the efficiency and optimization. Isac’s cones represent the largest class of convex cones discovered till now in separated locally convex spaces ensuring the existence and important properties for the efficient points under completeness instead of compactness.


INTRODUCTION
This is a very short research work representing an homage to Professor George Isac, Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. 17000, Kingston, Ontario, Canada, K7K 7B4. Professor Isac introduced the notion of "nuclear cone" in [1], published in [2] and called later on "supernormal cone" since it appears stronger than the usual concept of "normal cone". For the first time, we named these convex cones as "Isac'c Cones" in [3], after the acceptance on professor Isac's part. This study is devoted to Isac's cones, including significant examples, comments and several pertinent references, with the remark that this notion is more interesting in the Hausdorff locally convex spaces as in the normed linear spaces, having strong implications and applications in the efficiency and the optimization. A generalization of Isac's cones in the general Vector Spaces was given by us in [4].

ISAC'S (NUCLEAR OR SUPER-NORMAL) CONES
Throughout the research works devoted to nuclear (supernormal) cones professor Isac considered any locally convex space in the sense of the next definition. Definition 1. [5]. A locally convex space is any couple ( , ( )) X Spec X which is composed of a real linear space X and a family ( ) Spec X of seminoms on X such that: sup( , )( ) sup( ( ), ( )), p p x p x p x x X = ∀ ∈ It is well known [5] that whenever such a family as this ( ) Spec X is given on a real vector space where θ is the null vector in X . In this research paper we will suppose that the space ( , ) X τ sometimes denoted by X is a Hausdorff locally convex space. Every non-empty subset K of X satisfying the following properties: If, in addition, , then K is called pointed. Clearly, any pointed convex cone K in X generates an ordering on X defined by ( , ) and its corresponding polar is It is well known that the concept of normal cone is the most important notion in the theory and applications of convex cones in topological ordered vector spaces. Thus, for example, for every separated locally convex space ( , ( )) X Spec X and any closed normal cone ( , [6,7]). Each pointed convex cone ( , ( )) K X Spec X ⊂ for which there exists a non-empty, convex bounded set Remark 1. For the first time, we called any such as this cone "Isac's cone" in [4], taking into account that the above definition of locally convex spaces is equivalent with the following: Let X be a real or complex linear space and has the properties: Therefore, ς 0 (x) is a base of neighbourhoods for x and taking ς ( ) { : ∪ ∅ is the locally convex topology generated by the family P .
Obviously, the usual operations which induce the structure of linear space on X are continuous with respect to this topology. The corresponding topological space ( , ) The best, special, refined and non -trivial Isac's cones classes associated to the sets of all normal cones in arbitrary Hausdorff locally convex spaces was introduced and studied in [9] as the full nuclear cones, these families of convex cones being defined as follows: if ( , ( )) X Spec X is an arbitrary locally convex space, is a Hausdorff base of ( ) Spec X and K X ⊂ is a normal cone, then for any mapping

R) is not an Isac's cone if p>1, that is, it is not well-based in all these cases.
However, these cones are normal for every p ≥.
The same conclusion concerning the nonsupernormality is valid for the positive orthant of any usual Orlicz space. 9. In l p (p≥1) equipped with the usual norm ⋅ p the positive cone is also normal with respect to its usual norm topology, but it is not an Isac's cone excepting the case p = 1. Indeed, for every p>1, the sequence (e n ) having 1 at the nth coordinate and zeros elsewhere converges to zero in the weak topology, but not in the norm topology and by virtue of Theorem 1 it follows that C p is not an Isac's cone. For p = 1, C p is well-based by the set B = and Proposition 5 given in [2] ensures that it is an Isac's cone. If we consider in this case the locally convex topology in l 1 converges weakly to zero, but it is not convergent to zero in the H-locally convex topology. The results follows by Theorem 1. It is clear that every weak topology is a H-locally convex topology and, in all these cases, the supernormality of the convex cones coincides with the normality thanks to the Corollary of the Proposition 2 given in [2]. 10. In the space C([a, b] The hypothesis that all x∈K are concave is essential for the supernormality. 11. The convex cone of all nonnegative sequences in the space of all absolutely convergent sequences is the dual of the usual positive cone in the space of all convergent sequences. Consequently, it has a weak star compact base and, hence, it is a weak star supernormal cone. 12. In l ∞ or in c 0 equipped with the supremum norm, the convex cone consisting of all sequences having all partial sums nonnegative is not normal, so it is not supernormal. 13. In every Hausdorff locally convex space any normal cone is supernormal with respect to the weak topology. 14. In every locally convex lattice which is a (L)-space the ordering cone is supernormal (see also the Example 7 given in [11] [30]. Such a convex cone was initially called "nuclear cone" by Isac, G. (1981) because in every nuclear space (Pietch, A., 1972) (Isac.G., 2003;Isac, G., Tammer, Chr., 2003). Therefore, the more appropriate background for Isac's cones is any separated locally convex space.

CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS
The family of Isac's cones represents the largest class of ordering cones in Hausdorff locally convex spaces ensuring the existence and the adequate properties for the efficient points sets involved in the general optimization, following different completeness types instead of compactness. Consequently, one of the main goal of the next research is to identify new applications of Isac's cones in the efficiency projected in the best approximation problems, the set -valuet fixed point theory including the dynamical systems and the nuclearity of the linear vector spaces.