Convex Functions, Monotone Operators and Differentiability
Convex Functions, Monotone Operators and Differentiability
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DOI：
10.1007/978-3-662-21569-2
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发表时间：
1989-02
影响因子：
5.2
通讯作者：
R. Phelps
中科院分区：
文献类型：
--
作者：
R. Phelps
改进和扩充的第二版包含了第一版以来几年来取得的一些主要成果的论述。Preiss肯定地回答了几十年前的一个问题：具有等价Gateaux可微范数的Banach空间是否为弱Asplund空间。罗卡菲勒关于凸函数次微分的极大单调性基本定理的西蒙斯极其简单的证明。戈德弗罗伊、德维尔和齐兹勒提出的有用的Borwein-Preiss光滑变分原理的激动人心的新版本。上过泛函分析课程的学生都可以使用这些材料；事实上，第一版已经在许多研究生研讨会上使用过。从直线上的凸函数出发，引出了Banach空间中凸函数的凸性、可微性和次可微性、单调算子的一般连续性、Banach空间的几何以及Radon-Nikodym性质、凸分析、变分原理和摄动优化等相互关联的主题。虽然其中大部分是经典的，但最近发现的简化证明在许多情况下都给出了。有许多练习，其中许多是博览会不可分割的一部分。
The improved and expanded second edition contains expositions of some major results which have been obtained in the years since the 1st edition. Theaffirmative answer by Preiss of the decades old question of whether a Banachspace with an equivalent Gateaux differentiable norm is a weak Asplund space. The startlingly simple proof by Simons of Rockafellar's fundamental maximal monotonicity theorem for subdifferentials of convex functions. The exciting new version of the useful Borwein-Preiss smooth variational principle due to Godefroy, Deville and Zizler. The material is accessible to students who have had a course in Functional Analysis; indeed, the first edition has been used in numerous graduate seminars. Starting with convex functions on the line, it leads to interconnected topics in convexity, differentiability and subdifferentiability of convex functions in Banach spaces, generic continuity of monotone operators, geometry of Banach spaces and the Radon-Nikodym property, convex analysis, variational principles and perturbed optimization. While much of this is classical, streamlined proofs found more recently are given in many instances. There are numerous exercises, many of which form an integral part of the exposition.