Conditional Analysis on R d
Conditional Analysis on R d
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DOI：
10.1007/978-3-662-48670-2_6
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发表时间：
2012-11
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影响因子：
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通讯作者：
Patrick Cheridito;Michael Kupper;Nicolas Vogelpoth
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文献类型：
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作者：
Patrick Cheridito;Michael Kupper;Nicolas Vogelpoth
本文提供了来自线性代数、实数分析和凸分析的经典结果的版本，这些结果是在 𝜎 有限测度空间上的可测函数环 𝐿0 上的有限秩自由模块中得到的。我们研究子模块是否是有限生成的问题，并引入更一般的概念：𝐿0 -仿射集、𝐿0 -凸集、𝐿0 -凸锥、𝐿0 -超平面和𝐿0 -半空间。我们研究正交补、正交分解和正交基的存在性。我们还研究 𝐿0 -线性、𝐿0 -仿射、𝐿0 -凸和𝐿0 -次线性函数，并引入连续性、可微性、方向导数和次梯度的概念。我们使用 Bolzano–Weierstrass 定理的条件版本来证明条件柯西序列收敛，并给出条件优化问题具有最优解的条件。我们证明了 𝐿0 超平面分离 𝐿0 凸集的结果，并研究了 𝐿0 凸共轭函数。我们提供了关于 𝐿0 凸函数的 𝐿0 次梯度存在性的结果，证明了 Fenchel–Moreau 定理的条件版本并研究了条件 inf 卷积。
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.