Conditional Analysis on R^d

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CHERIDITO, Patrick, Michael KUPPER, Nicolas VOGELPOTH, 2014. Conditional Analysis on R^d

@unpublished{Cheridito2014Condi-30395, title={Conditional Analysis on R^d}, year={2014}, author={Cheridito, Patrick and Kupper, Michael and Vogelpoth, Nicolas} }

Kupper, Michael Cheridito, Patrick eng 2015-03-18T13:17:43Z Conditional Analysis on R^d Cheridito, Patrick Kupper, Michael Vogelpoth, Nicolas Vogelpoth, Nicolas 2015-03-18T13:17:43Z 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 L<sup>0</sup> 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 L<sup>0</sup>-affine sets, L<sup>0</sup>-convex sets, L<sup>0</sup>-convex cones, L<sup>0</sup>-hyperplanes, L<sup>0</sup>-half-spaces and L<sup>0</sup>-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L<sup>0</sup>-linear, L<sup>0</sup>-affine, L<sup>0</sup>-convex and L<sup>0</sup>-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 L<sup>0</sup>-convex sets by L<sup>0</sup>-hyperplanes and study L<sup>0</sup>-convex conjugate functions. We provide a result on the existence of L<sup>0</sup>-subgradients of L<sup>0</sup>-convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional inf-convolutions. 2014

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