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- W2922081714 abstract "For each countable ordinal $alpha$ let $mathcal{S}_{alpha}$ be the Schreier set of order $alpha$ and $X_{mathcal{S}_alpha}$ be the corresponding Schreier space of order $alpha$. In this paper we prove several new properties of these spaces. 1) If $alpha$ is non-zero then $X_{mathcal{S}_alpha}$ possesses the $lambda$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $alpha$ is non-zero and $1<p<infty$ then the $p$-convexification $X^{p}_{mathcal{S}_alpha}$ possesses the uniform $lambda$-property of R. Aron and R. Lohman. 3) For each countable ordinal $alpha$ the space $X^*_{mathcal{S}_alpha}$ has the $lambda$-property. 4) For $nin mathbb{N}$, 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 $alpha$, $X_{mathcal{S}_alpha}$ is $(V)$-polyhedral and has the $lambda$-property implies that each $X_{mathcal{S}_alpha}$ 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 $c_0$." @default.
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- W2922081714 date "2019-03-08" @default.
- W2922081714 modified "2023-10-17" @default.
- W2922081714 title "On the geometry of higher order Schreier spaces" @default.
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