Formalization of Non-Archimedean Functional Analysis 1: Spherically Complete Spaces

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 C p Cₚ bold upper C Subscript 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.