Formalized functional analysis with semilinear maps

TL;DR

This paper formalizes semilinear maps in Lean's mathlib, unifying linear and conjugate-linear maps, and proves key theorems in functional analysis and algebraic classification over fields.

Contribution

It introduces a formalization of semilinear maps in proof assistant Lean, enabling the proof of fundamental theorems in functional analysis and algebraic classification.

Findings

Formalization of semilinear maps in Lean's mathlib

Proof of the Fréchet–Riesz and spectral theorems for semilinear maps

Application to classification of isocrystals over algebraically closed fields

Abstract

Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean's \textsf{mathlib} library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fr\'echet--Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonn\'e and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic.