AI 中文总结
本文构造了两个可分实巴拿赫空间，使得对任意小误差存在标准精确$\varepsilon$-等距映射，但不存在等距嵌入，从而否定回答了Cheng和Zhou的开放问题。
AI 中文摘要
我们证明存在两个可分实巴拿赫空间$X$和$Y$，使得对任意$\varepsilon>0$，存在一个标准精确$\varepsilon$-等距映射$f_\varepsilon: X\to Y$，其每个距离的畸变介于$0$和$\varepsilon$之间，然而$X$到$Y$的等距嵌入不存在，即使不假设线性性也是如此。这给出了Cheng和Zhou的\cite[问题1和问题2]{cheng4}的否定答案。该构造结合了雪花Lipschitz自由空间、Kalton的Schur性质定理以及Godefroy和Kalton的等距线性化定理。
英文摘要
We show that there exist two separable real Banach spaces $X$ and $Y$ such that, for every $\varepsilon>0$, there is a standard exact $\varepsilon$-isometry $f_\varepsilon: X\to Y$ whose distortion of every distance is between $0$ and $\varepsilon$, whereas no isometric embedding of $X$ into $Y$ exists, even without a linearity assumption. This gives a negative answer to \cite[Problem 1 and Problem 2]{cheng4} of Cheng and Zhou. The construction combines a snowflaked Lipschitz-free space with the Schur-property theorem of Kalton and the isometric linearization theorem of Godefroy and Kalton.
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