Formalization of non-Archimedean functional analysis 1: spherically complete spaces

Abstract: In this article, we present a formalization of spherically complete spaces, which is a fundamental notion in non-archimedean functional analysis. This work includes the equivalent definitions of spherically complete spaces, their basic properties, examples and non-examples such as the field $\mathbf{C}_p$ of $p$-adic complex numbers. As applications, we formalize the Birkhoff-James orthogonality, Hahn-Banach extension theorem and the spherical completion for non-archimedean Banach spaces. Code available at https://github.com/YijunYuan/SphericalCompleteness