关于$\ell_\infty(Γ)$的一致微分映射
On uniformly differentiable mappings from $\ell_\infty(Γ)$
摘要：
在1970年Haskell Rosenthal证明了，如果$X$是一个Banach空间，$Γ$是一个无限指数集，$T:\ell_\infty(Γ)\to X$是一个有界线性算子，使$\inf_{γ\inΓ}\|T(e_γ)\|>0$，然后$T$作为一个同构的$\ell_\infty(Γ')$，为一些$Γ'\subsetΓ$相同的基数作为$Γ$。我们的主要结果是这个定理的非线性加强。更精确地说，在GCH和$Γ$的正则性的假设下，我们证明了如果${F}:B_{\ell_\infty(Γ)}\to X$是一致可微的，并且使得$\inf_{γ\inΓ}\|{F}(e_γ){-F(0)}\|>0$，那么存在$x\in B_{\ell_\infty(Γ)}$，使得$d{F}(x)[\cdot]$是一个有界线性算子，它充当$\ell_\infty(Γ')$上的同构，对于某些与$Γ$具有相同基数的$Γ'\subsetΓ$。
英文摘要：
In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $Γ$ is an infinite index set, and $T:\ell_\infty(Γ)\to X$ is a bounded linear operator such that $\inf_{γ\inΓ}\|T(e_γ)\|>0$ then $T$ acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of $Γ$, we show that if ${F}:B_{\ell_\infty(Γ)}\to X$ is uniformly differentiable and such that $\inf_{γ\inΓ}\|{F}(e_γ){-F(0)}\|>0$ then there exists $x\in B_{\ell_\infty(Γ)}$ such that $d{F}(x)[\cdot]$ is a bounded linear operator which acts as an isomorphism on $\ell_\infty(Γ')$, for some $Γ'\subsetΓ$ of the same cardinality as $Γ$.
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