On a question of Pietch

TL;DR

This paper characterizes when finite-dimensional normed spaces embed isometrically into _p spaces, linking it to discrete Levy p-representations, and explores implications for subspace isometries and smoothness properties.

Contribution

It provides a new characterization of isometric embeddings into _p spaces via Levy p-representations and clarifies restrictions on subspace isometries, especially for _2 and _q spaces.

Findings

Finite-dimensional normed spaces embed isometrically into _p if and only if they have a discrete Levy p-representation.

_2^2 is not isometric to a subspace of _p unless p is an even integer.

_q^2 is not isometric to a subspace of _p unless q=p for q_2 and q_p.

Abstract

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 $q\neq 2$ turns out to be much more restrictive. The main result combined with a result of Dor provides a proof of the fact that if $q\neq 2$ 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.