Real and complex linear extensions for locally convex cones
Locally convex (preordered) cones were introduced by the author and \textit{K. Keimel} [``Ordered cones and approximation'', Lect. Notes Math. 1517 (1992; Zbl 0752.41033)] in order to have a unifying concept, in which Korovkin-type theorems can be treated, in their definition of a cone there is no cancellation law (i.e. \(a+c= b+c\) implies \(a=b\)), therefore there is no embedding into a real vector space in general. A locally convex cone \(P\) is called linear over \(\mathbb{K}\) \((\in\{\mathbb{R},\mathbb{C}\})\) if the operation of \(\mathbb{R}^+\) on \(P\) is extended to \(\mathbb{K}\), such that right-distributivity, associativity, and a weakened form of left-distributivity \(((\alpha+\beta)a\leq \alpha a+\beta a)\) is valid; if, furthermore, a certain condition concerning a modification of the given order holds, then \(P\) is called a locally convex cone over \(\mathbb{K}\). The first result of the paper shows, that every locally convex cone can be embedded into a locally convex cone over \(\mathbb{K}\). In the second part continuous linear functionals on these cones are investigated. After defining regular linear functionals a kind of integral representation theorem is proved. The theory is illustrated by the example formed by the set of non-empty closed convex sets of a locally convex ordered topological vector space.
locally convex ordered topological vector space
scientific article; zbMATH DE number 1117826
scientific article