A metric interpretation of reflexivity for Banach spaces

Abstract

We define two metrics $d_{1,\alpha }$ and $d_{\infty ,\alpha }$ on each Schreier family ${\mathcal{S}}_{\alpha }$, $\alpha <{\omega }_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha <{\omega }_1$, so that there is no mapping $\Phi :{\mathcal{S}}_{\alpha }\to X$ for which $$c{d}_{\infty ,\alpha }(A,B)\le \Vert \Phi (A)-\Phi (B)\Vert \le C{d}_{1,\alpha }(A,B)\text{ for all A,B∈Sα.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $\alpha$ that $\max (\text{ Sz}(X),\text{ Sz}(X^{\ast }))\le \alpha$ if and only if $({\mathcal{S}}_{\alpha },d_{1,\alpha })$ does not bi-Lipschitzly embed into $X$. Here $\text{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$.

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• $(\beta )$-distortion of some infinite graphs

Cited by in corpus (1)

• Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes