论文标题
Banach空间的波兰空间。等轴测和同构类别的复杂性
Polish spaces of Banach spaces. Complexity of isometry and isomorphism classes
论文作者
论文摘要
我们研究了作者[14]中最近引入和调查的Banach空间的波兰空间中可分离的Banach空间的等轴测和异态性类别的复杂性。我们获得了有关最古典可分开的Banach空间的明显结果。 我们证明，无限尺寸可分离的希尔伯特空间的特征是唯一可分开的无限尺寸Banach空间，其等轴测类别已关闭，也是独特的可分离无限尺寸的无限尺寸Banach空间，其同构型类别为$f_σ$。对于$ p \ in \ weft [1,2 \ right）\ cup \ left（2，\ infty \ right）$，我们表明$ l_p [0,1] $和$ \ ell_p $的等轴测类别分别是$g_Δ$ - complete sets和$ f_ {c {σδ} $ - 完整的集合。然后，我们证明$ C_0 $的等轴测类是$ f_ {σδ} $ - 完整集。 此外，我们计算了许多其他自然类别可分开的Banach空间的复杂性。例如，可分离$ \ Mathcal {l} _ {p，λ+} $ - 对于$ p，λ\ geq 1 $的空间，被证明是$g_Δ$ - set，超级反复空间类别显示为$ f_ {n = n = f _ {σδ} $ - $ - $ \boldsymbolς^0_6 $ -SET。本文结束了许多开放问题和未来研究的建议。
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 $p\in\left[1,2\right)\cup\left(2,\infty\right)$, we show that the isometry classes of $L_p[0,1]$ and $\ell_p$ are $G_δ$-complete sets and $F_{σδ}$-complete sets, respectively. Then we show that the isometry class of $c_0$ 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 $\mathcal{L}_{p,λ+}$-spaces, for $p,λ\geq 1$, 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 $\boldsymbolΣ^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.
