Generic differentiability of Lipschitzian functions
Generic differentiability of Lipschitzian functions
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Lipschitzian 函数的一般可微分
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发表时间：
1979
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通讯作者：
G. Lebourg
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作者：
G. Lebourg
它表明，在可分的拓扑向量空间，这是Baire空间，通常的性质，已被引入研究的本地“一阶”行为的实值函数满足Lipschitz型条件是“一般”等价的，从而导致一个独特的类“一般光滑”的功能。这些功能的特点切锥和方向导数和他们的“一般”的可微性进行了研究。所得结果推广了连续凸函数的一些著名的可微性。导论.研究了定义域为拓扑向量空间开子集的局部Lipschitz实值函数的通有光滑性。最近在凸函数的限制情况下这个问题又重新出现了[2]，[3]，[9]，[10]，[14];沿着这条脉络的第一个结果似乎是由于E。Asplund和断言，每一个连续凸函数定义在一个开放的凸子集的Banach空间是Gateaux（甚至Frechet）可微的一个密集的G8子集的域，提供了存在的对偶规范，享有一些不错的几何性质。不幸的是，众所周知的例子表明，这样的性质不再适用于任意的局部Lipschitz实值函数。因此，许多作者试图推广经典的Rademacher定理，引入理论上的“测度零”集的新概念[1]，[6]，[13]。然而，大量的工作可以做，以扩大通用的可微性性质的一大类实值函数。这篇笔记的主要贡献是至少在“小”空间（这里指的是可分空间）中给出了这些函数的特征，用方向导数和切锥表示。第一节讨论了由F. H. [7][8]我从他们那里得到了很多好处。定理（1.7）是我们发展的基石，它允许我们在定理（2.1）和（2.2）的证明中免除凸性假设。在下一节中，我们将讨论局部光滑性质之间的关系，编辑于1978年4月27日收到。AMS（MOS）主题分类（1970年）。小学58 C20;中学26 A24、26 A27。（1979 American Mathematical Society 0002-9947/79/0000-0555 /$06.00 125本内容于2016年6月22日星期三05：22：12 UTC从207.46.13.124下载所有使用受 http://about.jstor.org/terms约束
It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local "first order" behaviour of real-valued functions which satisfy a Lipschitz type condition are "generically" equivalent and thus lead to a unique class of "generically smooth" functions. These functions are characterized in terms of tangent cones and directional derivatives and their "generic" differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions. Introduction. This paper is devoted to the investigation of generic smoothness properties for locally Lipschitzian real-valued functions, the domain of which is an open subset of a topological vector space. There has been a recent revival of this problem in the restricted case of convex functions [2], [3], [9], [10], [14]; the first result along this vein seems to be due to E. Asplund and asserts that every continuous convex function defined on an open convex subset of a Banach space is Gateaux (or even Frechet) differentiable on a dense G8 subset of its domain, provided the existence of a dual norm which enjoys some nice geometrical properties. Unfortunately, well-known examples show that such a property no longer holds for an arbitrary locally Lipschitzian real-valued function. Therefore, many authors have sought a generalization of the classical Rademacher theorem involving a new concept of theoric "measure-zero" set [1], [6], [13]. However, a lot of work can be done to extend the generic differentiability properties to a large class of real-valued functions. The main contribution of this note is to give, at least in "small" spaces (here it means separable spaces), a characterization of these funcions, expressed both in terms of directional derivatives and tangent cones. The first section deals with the notion of a generalized gradient introduced by F. H. Clarke, [7], [8], from whom I have benefitted greatly. Theorem (1.7) is the cornerstone to our development and allows us to dispense with convexity assumptions in the proofs of Theorems (2.1) and (2.2). In the next section, we discuss the relation between smoothness properties of locally Received by the editors April 27, 1978. AMS (MOS) subject classifications (1970). Primary 58C20; Secondary 26A24, 26A27. ( 1979 American Mathematical Society 0002-9947/79/0000-0555 /$06.00 125 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:22:12 UTC All use subject to http://about.jstor.org/terms