The main result is that a finite dimensional normed space embeds isometrically in \(\ell _p\) if and only if it has a discrete Levy p-representation. This provides an alternative answer to a question raised by Pietch, and as a corollary, a simple proof of the fact that unless p is an even integer, the two-dimensional Hilbert space \(\ell _2^2\) is not isometric to a subspace of \(\ell _p\). The situation for \(\ell _q^2\) with \(q\ne 2\) turns out to be much more restrictive. The main result combined with a result of Dor provides a proof of the fact that if \(q\ne 2\) then \(\ell _q^2\) is not isometric to a subspace of \(\ell _p\) unless \(q=p\). Further applications concerning restrictions on the degree of smoothness of finite dimensional subspaces of \(\ell _p\) are included as well.

主要结果是，当且仅当它具有离散的Levy p-表示时，有限维范数空间才等距地嵌入\（\ ell _p \）。这为Pietch提出的问题提供了另一种答案，作为推论，一个简单的事实证明，除非p是偶数整数，否则二维希尔伯特空间\（\ ell _2 ^ 2 \）并非等距\（\ ell _p \）的子空间。的局面\（\ ELL _q ^ 2 \）与\（Q \ NE 2 \）原来是更限制性的。主结果与Dor的结果相结合提供了以下事实的证明：如果\（q \ ne 2 \）则\（\ ell _q ^ 2 \）除非\（q = p \）相对于\（\ ell _p \）的子空间不是等距的。还包括关于限制\（\ ell _p \）的有限维子空间的平滑度的其他应用。