复域上的巴拿赫等距猜想
Banach's Isometric Conjecture over the Complex Field
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中文总结 AI 辅助
该研究完成复域巴拿赫等距猜想证明，还证四元数域对应结论，用Lu和Yang的丛次数机制完成核心几何论证，且将结果推广到多类空间与度量。
中文摘要 AI 辅助
我们完成了复域上的巴拿赫等距猜想。更准确地说，若\boldsymbol{X}是一个复赋范空间，且对于某个满足\boldsymbol{2}\boldsymbol{\boldsymbol{n}}\boldsymbol{\boldsymbol{<}}\boldsymbol{\boldsymbol{dim}}_{\boldsymbol{\boldsymbol{\boldsymbol{C}}}}\boldsymbol{\boldsymbol{X}}的整数\boldsymbol{n}，其所有\boldsymbol{n}维复子空间作为度量空间是等距的，则该范数由埃尔米特内积诱导。我们还证明了四元数域对应的结论。核心几何论证首先处理不具有凸性或中心对称性的实星体；将其应用于圆复体或四元数体时，可得出相互实线性等价的超平面截面迫使环境体为埃尔米特椭球的结论。该证明采用了Lu和Yang针对实情形引入的丛次数机制。最后，我们将结果推广到绝对齐次函数、分次弗雷歇空间、可度量化局部凸空间以及相容平移不变度量。
英文摘要
We complete Banach's isometric conjecture over the complex field. More precisely, if \(X\) is a complex normed space and, for some \(2\leqslant n<\dim_{\C}X\), all its \(n\)-dimensional complex subspaces are isometric as metric spaces, then the norm is induced by a Hermitian inner product. We also prove the quaternionic counterpart. The central geometric argument first treats real star bodies without convexity or central symmetry; applied to circled complex or quaternionic bodies, it shows that mutually real-linearly equivalent hyperplane sections force the ambient body to be a Hermitian ellipsoid. The proof adapts the bundle-degree mechanism introduced by Lu and Yang for the real case. Finally, we obtain extensions to absolutely homogeneous functions, graded Fréchet spaces, metrisable locally convex spaces, and compatible translation-invariant metrics.
