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.