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

Abstract

In this article, we present a formalization of spherically complete spaces, a fundamental notion in non-Archimedean functional analysis, using the Lean theorem prover (v4.31.0), building over Mathlib. 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 notion of Birkhoff-James orthogonality, the Hahn-Banach extension theorem and the spherical completion for non-Archimedean Banach spaces. URL of code: https://github.com/YijunYuan/SphericalCompleteness/tree/paper

Final version. Accepted by Journal of Symbolic Logic