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

2022/04/14 by Cúth, Marek, Doležal, Martin, Doucha, Michal +1 · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · doi:10.48550/arxiv.2204.06834

Abstract

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is F_σ. For p∈[1,2)∪(2,∞), we show that the isometry classes of L_p[0,1] and ℓ_p are G_δ-complete sets and F_σδ-complete sets, respectively. Then we show that the isometry class of c₀ is an F_σδ-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable L_p,λ+-spaces, for p,λ≥ 1, is shown to be a G_δ-set, the class of superreflexive spaces is shown to be an F_σδ-set, and the class of spaces with local Π-basis structure is shown to be a \boldsymbolΣ⁰₆-set. The paper is concluded with many open problems and suggestions for a future research.

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