Symmetric finite representability of $\ell^p$ in Orlicz spaces

S. V. Astashkin

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 4, pp. 15-24

Résumé

It is well known that a Banach space need not contain any subspace isomorphic to a space $\ell^p$ $(1\le p\infty)$ or $c^0$ (it was shown by Tsirel'son in 1974). At the same time, by the famous Krivine's theorem, every Banach space $X$ always contains at least one of these spaces locally, i.e., there exist finite-dimensional subspaces of $X$ of arbitrarily large dimension $n$ which are isomorphic (uniformly) to $\ell_p^n$ for some $1\le p\infty$ or $c_0^n$. In this case one says that $\ell^p$ (resp. $c^0$) is finitely representable in $X$. The main purpose of this paper is to give a characterization (with a complete proof) of the set of $p$ such that $\ell^p$ is symmetrically finitely representable in a separable Orlicz space.

Détail

Keywords: $\ell^p$-space, finite representability of $\ell^p$-spaces, symmetric finite representability of $\ell^p$-spaces, Orlicz function space, Orlicz sequence space, Matuszewska-Orlicz indices.