Let K be the scalar field of the first orthomodular (or Form Hilbert) space, described by H. Keller in 1980. It has a non-Archimedean order, an infinite rank valuation compatible with the order as well as an explicitly defined ultrametric, all of which induce the same topology. We study analytic functions defined on valued field K, and we will establish an invertibility local theorem for these functions as an application of Banach fixed point theorem.

中文翻译：

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具有无穷秩估值的有序域中的解析函数

令 K 为第一个正模（或形式希尔伯特）空间的标量场，由 H. Keller 在 1980 年描述。它具有非阿基米德阶、与阶兼容的无限秩估值以及明确定义的超度量，所有其中诱导相同的拓扑结构。我们研究了定义在值域 K 上的解析函数，我们将建立这些函数的可逆局部定理作为 Banach 不动点定理的应用。