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

2026-01-29

AI Summary

This work presents the first complete formalization of non-Archimedean functional analysis within an interactive theorem prover, addressing the long-standing lack of a rigorous formal foundation for spherically complete spaces. It provides a precise definition of spherical completeness and formally verifies its various equivalent characterizations, fundamental properties, and both typical and atypical examples—including a machine-checked proof that $C_p$ is not spherically complete. Building on this foundation, the study further formalizes key concepts such as Birkhoff–James orthogonality, the Hahn–Banach extension theorem, and the spherical completion process for non-Archimedean Banach spaces. This contribution establishes the first machine-verified formal framework for non-Archimedean analysis, filling a critical gap in the landscape of formalized mathematics.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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

Problem

• non-Archimedean functional analysis
• spherically complete spaces
• formalization
• Banach spaces
• p-adic complex numbers

Innovation

• spherically complete spaces
• non-Archimedean functional analysis
• formalization
• Hahn-Banach theorem
• Birkhoff-James orthogonality