发表机构
Indiana University Bloomington（印第安纳大学布卢明顿分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对不包含$C(2^\mathbb{N})$同构副本的可分巴拿赫空间，研究其非通用树的阶，证明可分希尔伯特空间相对于Haar基的非通用树阶为$\omega + 1$，是该度量下复杂性最低的类。
AI 中文摘要
Bossard在其1994年的博士论文中，为每个不包含同构副本$C(2^\mathbb{N})$的可分巴拿赫空间$X$引入了非通用树$T_{NU}(X)$的概念。结合在良基树上定义的阶运算，我们获得了一种通过$C(2^\mathbb{N})$有限维子空间的同构度来对可分巴拿赫空间的复杂性进行分类的方法。尽管该概念在后续年份中得到了进一步完善，但目前尚无针对具体空间和基直接展示非通用树阶的实例。本文证明，若$H$为任意可分希尔伯特空间，则相对于$C(2^\mathbb{N})$的Haar基，$o(T_{NU}(H)) = \omega + 1$，这表明在考虑该基时，希尔伯特空间类是该度量下复杂性最低的类。
英文摘要
In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree $T_{NU}(X)$ associated with each separable Banach space $X$ which does not contain an isomorphic copy of $C(2^\mathbb{N})$. Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of $C(2^\mathbb{N})$. Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if $H$ is any separable Hilbert space, then $o(T_{NU}(H)) = ω + 1$ when taken with respect to the Haar basis for $C(2^\mathbb{N})$, demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.
Comments9 pages
