论文标题
通过局部特征可怜性的特征可怜性
Frechet differentiability via partial Frechet differentiability
论文作者
论文摘要
令$ x_1，\ dots，x_n $为banach空间，$ f $ a $ x = x_1 \ times \ dots \ times \ times x_n $上的真实功能。令$ a_f $为x $的所有点$ x \ in $ f $部分是fr \'echet可区分的集合，但不是fr \'echet可区分。我们的结果表明，如果$ x_1，\ dots，x_ {n-1} $是asplund空间，而$ f $在$ x $上是连续的（resp。lipschitz），那么$ a_f $是第一个类别集（resp。A$σ$ -UPER -UPPER PORFOR SET）。我们还证明，如果$ x $，$ y $是可分离的巴纳克空间，$ f：x \ to y $是lipschitz的映射，那么所有点$ x \ in x $ in x $的集合是$ f $是g \^ ateaux aTeaux可区分的，fr \ echet echet echet可在有限的codimention of Firite codimention of Firite codimenspace上是$ fr fr fr fr fr fr fr \'eChet eChet eChet eChet eChet。还证明了许多相关的更一般结果。
Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\' echet differentiable along a closed subspace of finite codimension but is not Fr\' echet differentiable, is $σ$-upper porous. A number of related more general results are also proved.
