arXiv2019
For each countable ordinal $α$ let $\mathcal{S}_α$ be the Schreier set of order $α$ and $X_{\mathcal{S}_α}$ be the corresponding Schreier space of order $α$. In this paper we prove several new properties of these spaces. 1) If $α$ is non-zero then $X_{\mathcal{S}_α}$ possesses the $λ$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $α$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}_{\mathcal{S}_α}$ possesses the uniform $λ$-property of R. Aron and R. Lohman. 3) For each countable ordinal $α$ the space $X^*_{\mathcal{S}_α}$ has the $λ$-property. 4) For $n\in \mathbb{N}$, if $U:X_{\mathcal{S}_n}\to X_{\mathcal{S}_n}$ is an onto linear isometry then $Ue_i = \pm e_i$ for each $i \in \mathbb{N}$. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for non-zero $α$, $X_{\mathcal{S}_α}$ is $(V)$-polyhedral and has the $λ$-property implies that each $X_{\mathcal{S}_α}$ is an example of space solving a problem of J. Lindenstrauss from 1966. The first example of such a space was given by C. De Bernardi in 2017 using a renorming of $c_0$.