Separation Theorem With Respect to Sub-Topical Functions and Abstract Convexity

Abstract

Abstract. This paper deals with topical and sub-topical functions in a class of ordered Banach spaces. The separation theorem for downward sets and sub-topical functions is given. It is established some best approximation problems by sub-topical functions and we will characterize sub-topical functions as superimum of elementary sub-topical functions.

Topical functions on semimodules and generalizations

“…in which every subset has a supremum and an infimum, or in a Banach space X, the implication (43) is true, since in these cases the proof of (43) given in [9, Proposition 2] and [1, Theorem 2.5] works (this is the underlying fact that has permitted in [10,11] and [9,1] to extend some results on functions with values in R to functions with values in R max (and even in R = R max ∪ {+∞}). However, it does not work in R n max , which is only b-complete, but not complete, which explains the counter-example given in Remark 5(b) above.…”

“…More generally, such mappings as (51), defined with the aid of a "coupling function" ϕ(x, y), have been introduced without any name in [10, Theorem 8.3 and Remark 8.3], for X = R I and K = R, where I is an arbitrary index set, in [11,Theorem 9.4 ] for X = A n and K = A, with A being a conditionally complete lattice ordered group, and in [1] for X belonging to "a class of ordered Banach spaces"; however, the latter X can be identified linearly and latticially with C(S) ⊂ R S , where S is a compact Hausdorff space and C(S) is the Banach space of all real-valued continuous functions on S (see the Introduction).…”

“…The converse of Theorem 14 is not valid, i.e., not every pointwise sup of quasi-elementary affine functions is strongly supertopical. Indeed, for example in the particular case where X = R n max and K = R max , in [10,Theorem 8.3] it has been shown that the representation (52) holds even for the more general class of the subtopical functions; for a related result see also [1,Theorem 3.3].…”

“…Some authors, e.g. Mohebi [9], Alimohammady and Shahmari [1] and others, have observed that the following concept is more general: X is a partially ordered Banach space whose positive cone {x ∈ X|x 0} contains an interior point u such that…”