The Complex Banach Isometric Conjecture

Banach asked whether a normed space must be Hilbert if, for one fixed dimension greater than one, all subspaces of that dimension are linearly isometric. We prove the complex case. By the codimension-one reduction, the essential finite-dimensional problem is to characterize a balanced convex body in a complex space whose complex hyperplane sections are all complex-linearly equivalent. We show that such a body is a Hermitian ellipsoid. The proof adapts the recent bundle-degree method of Lu and Yang for the real problem to the complex setting. After covariance normalization, the complex-linear symmetry group of a model section gives rise to a principal bundle over an odd-dimensional sphere. Passing to the identity component shows that the relevant obstruction lies in a finite homotopy group. A suitable positive-degree pullback therefore trivializes the bundle and produces a global Lipschitz family of exact complex-linear section maps. Brouwer degree and a signed degree formula then imply strong polynomial constraints on the norm of the model section. Unique factorization forces its square to be quadratic, while phase invariance makes the resulting quadratic form Hermitian. The parallelogram identity then yields the general complex Banach-space statement. Submission history This manuscript was submitted to arXiv on August 15, 2026 (temporary submission number submit/7952332). As of the creation of this Zenodo record, the arXiv submission dashboard shows the manuscript as under moderation (“on hold”). This Zenodo record also includes supporting documentation consisting of: a screenshot of the arXiv submission page showing submit/7952332 in “on hold” status; and a screenshot of the automated arXiv submission-receipt email associated with the August 15, 2026 submission. A separate manuscript addressing the same problem, Antonio Acuaviva and Tomasz Kania, “Banach's Isometric Conjecture over the Complex Field,” arXiv:2608.18257, was submitted to arXiv on August 18, 2026 and became publicly available while the present arXiv submission was still shown as “on hold.”