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

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 $F_\sigma$. 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 $G_\delta$-complete sets and $F_{\sigma\delta}$-complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\sigma\delta}$-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\mathcal{L}_{p,\lambda+}$-spaces, for $p,\lambda\geq 1$, is shown to be a $G_\delta$-set, the class of superreflexive spaces is shown to be an $F_{\sigma\delta}$-set, and the class of spaces with local $\Pi$-basis structure is shown to be a $\boldsymbol{\Sigma}^0_6$-set. The paper is concluded with many open problems and suggestions for a future research.