Subdifferentials in Boolean-valued models of set theory
In accordance with \textit{B. Z. Vulikh} [Introduction to the theory of partially ordered spaces (Russian) (1961; Zbl 0101.085); cf. also the English translation (1967; Zbl 0186.446)], a lattice ordered vector space satisfying the Dedekind principle (every bounded non empty set has an infimum and supremum) is called a K-space. If X is a real vector space and Y is a K-space, then the subdifferential \(\partial P\) of a sublinear operator P:X\(\to Y\) is the set of all linear operators from X into Y dominated by P, i.e. the set of those \(A\in L(X,Y)\) for which A(x)\(\leq P(x)\) for each \(x\in X\). The basis B(Y) of the K-space Y consists of all units of Y (e is a unit iff \(\inf (e,1-e)=0)\). B(Y) is a complete Boolean algebra and determines the space Y in certain sense. The main idea used in the paper is based on a result of \textit{E. I. Gordon} [Dokl. Akad. Nauk SSSR 237, 773-775 (1977; Zbl 0406.03064)]: the set R of all Boolean functions of the Boolean universe \(V^{(B)}\) which are reals with value 1 is a K-space with the basis isomorphic to B. For a given K-space Y and a non-empty set A the canonical sublinear operator \({\mathcal E}_ A\) from \(\ell_{\infty}(A,Y)\) (bounded functions from A into Y) into Y is defined as \({\mathcal E}_ A(f)=\sup \{f(x)\); \(x\in A\}\). Using the Boolean universe \(V^{(B)}\) with \(B=B(Y)\) the extremal points of the subdifferential \(\partial {\mathcal E}_ A\) are investigated.
scientific article; zbMATH DE number 3914312
scientific article