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# Conditional Analysis on ℝ𝑑

## Author(s): Cheridito, Patrick; Kupper, Michael; Vogelpoth, Nicolas

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 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 and 𝐿0 -halfspaces. 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. Publication Date: 2015 Citation: Cheridito, Patrick, Michael Kupper, and Nicolas Vogelpoth. "Conditional Analysis on ℝ𝑑." In Set Optimization and Applications - The State of the Art. Springer Proceedings in Mathematics & Statistics 151, pp. 179-211. doi:10.1007/978-3-662-48670-2_6 DOI: 10.1007/978-3-662-48670-2_6 ISSN: 2194-1009 Pages: 179-211 Type of Material: Conference Article Series/Report no.: Springer Proceedings in Mathematics & Statistics; Journal/Proceeding Title: Set Optimization and Applications - The State of the Art. Version: Author's manuscript

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