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.
@InProceedings{dupuis_et_al:LIPIcs.ITP.2022.10, author = {Dupuis, Fr\'{e}d\'{e}ric and Lewis, Robert Y. and Macbeth, Heather}, title = {{Formalized functional analysis with semilinear maps}}, booktitle = {13th International Conference on Interactive Theorem Proving (ITP 2022)}, pages = {10:1--10:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-252-5}, ISSN = {1868-8969}, year = {2022}, volume = {237}, editor = {Andronick, June and de Moura, Leonardo}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.10}, URN = {urn:nbn:de:0030-drops-167191}, doi = {10.4230/LIPIcs.ITP.2022.10}, annote = {Keywords: Functional analysis, Lean, linear algebra, semilinear, Hilbert space} }
Feedback for Dagstuhl Publishing