Conditional Analysis on R^d

Abstract

This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring L0 of measurable functions on a σ-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of L0-affine sets, L0-convex sets, L0-convex cones, L0-hyperplanes, L0-half-spaces and L0-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L0-linear, L0-affine, L0-convex and L0-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano-Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of L0-convex sets by L0-hyperplanes and study L0-convex conjugate functions. We provide a result on the existence of L0-subgradients of L0-convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional inf-convolutions.