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oai:drops-oai.dagstuhl.de:16719 2024-03-06T09:58:08Z ddc:004 open_access

Formalized functional analysis with semilinear maps Dupuis, Frédéric Lewis, Robert Y. Macbeth, Heather Functional analysis Lean linear algebra semilinear Hilbert space 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 mathlib library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fréchet-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 Dieudonné and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Frédéric Dupuis and Robert Y. Lewis and Heather Macbeth 2022 Is Part Of LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITP.2022.10 urn:nbn:de:0030-drops-167191 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.10 eng https://creativecommons.org/licenses/by/4.0/legalcode