Formalized functional analysis with semilinear maps

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.