KOPS - The Institutional Repository of the University of Konstanz
# Conditional Analysis on R^d

Type of Publication: | Preprint |

Author: | Cheridito, Patrick; Kupper, Michael; Vogelpoth, Nicolas |

Year of publication: | 2014 |

ArXiv-ID: | arXiv:1211.0747v2 |

Summary: |
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
^{0} 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^{0}-affine sets, L^{0}-convex sets, L^{0}-convex cones, L^{0}-hyperplanes, L^{0}-half-spaces and L^{0}-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L^{0}-linear, L^{0}-affine, L^{0}-convex and L^{0}-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^{0}-convex sets by L^{0}-hyperplanes and study L^{0}-convex conjugate functions. We provide a result on the existence of L^{0}-subgradients of L^{0}-convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional inf-convolutions. |

Subject (DDC): | 510 Mathematics |

Bibliography of Konstanz: | Yes |

Files | Size | Format | View |
---|---|---|---|

There are no files associated with this item. |

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@unpublished{Cheridito2014Condi-30395, title={Conditional Analysis on R^d}, year={2014}, author={Cheridito, Patrick and Kupper, Michael and Vogelpoth, Nicolas} }