失真问题

摘要：

我们证明了希尔伯特空间是可畸变的，并且事实上是任意可畸变的。这意味着对于所有的lambda >1，存在一个等价的范数。|.|使得对l_2的所有无穷维子空间Y，存在Y中的x，y，||x||_2 =| |y||_2 =1但|x| > λ |y|。我们还证明了：如果X是具有无条件基的无限维巴拿赫空间，则X的单位球面与l_1的单位球面是一致同胚的当且仅当X不一致包含l_infty ^n。

英文摘要：

We prove that Hilbert space is distortable and, in fact, arbitrarily distortable. This means that for all lambda >1 there exists an equivalent norm |.| on l_2 such that for all infinite dimensional subspaces Y of l_2 there exist x,y in Y with ||x||_2 = ||y||_2 =1 yet |x| >lambda |y|. We also prove that if X is any infinite dimensional Banach space with an unconditional basis then the unit sphere of X and the unit sphere of l_1 are uniformly homeomorphic if and only if X does not contain l_infty^n's uniformly.

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