Two Unexpected Examples Concerning Differentiability of Lipschitz Functions on Banach Spaces
Two Unexpected Examples Concerning Differentiability of Lipschitz Functions on Banach Spaces
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关于巴纳赫空间上 Lipschitz 函数可微性的两个意外例子
DOI：
10.1007/978-3-0348-9090-8_18
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
J. Tiser
中科院分区：
文献类型：
--
作者：
D. Preiss;J. Tiser
在这篇注记中，我们给出了两个例子，说明了关于Banach空间之间Lipschitz函数的可微性的两个主要概念的已知存在定理之间的一些令人惊讶的关系。回想一下，这些概念是：映射φ的G？teaux导数：X→Y at x∈X，它被定义为连续的线性映射φ‘(X)：X→Y验证
$$\Left\lang{\varphi‘(X)，u}\Right\Range=\mathop{\Lim}\Limits_{t\to 0}\frac{{\Phi\Left({x+tu}\Right)-\Phi(X)}}{t}$$
对于每个u∈X和Frechet导数，另外还要求以上极限对于∥u∥≤1是一致的。
In this note we present two examples illustrating some surprising relations between the known existence theorems concerning two main concepts of differentiability of Lipschitz functions between Banach spaces. Recall that these concepts are: Gâteaux derivative of a mapping φ: X → Y at x ∈ X, which is defined as a continuous linear map φ′ (x): X → Y verifying
$$ \left\langle {\varphi '(x),u} \right\rangle = \mathop {\lim }\limits_{t \to 0} \frac{{\phi \left( {x + tu} \right) - \phi (x)}}{t} $$
for every u ∈ X, and Frechet derivative which, in addition, requests that the above limit be uniform for ∥u∥ ≤ 1.