License:
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITP.2022.10
URN: urn:nbn:de:0030-drops-167191
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16719/
Dupuis, Frédéric ;
Lewis, Robert Y. ;
Macbeth, Heather
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 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.
BibTeX - Entry
@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/opus/volltexte/2022/16719},
URN = {urn:nbn:de:0030-drops-167191},
doi = {10.4230/LIPIcs.ITP.2022.10},
annote = {Keywords: Functional analysis, Lean, linear algebra, semilinear, Hilbert space}
}
Keywords: |
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Functional analysis, Lean, linear algebra, semilinear, Hilbert space |
Collection: |
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13th International Conference on Interactive Theorem Proving (ITP 2022) |
Issue Date: |
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2022 |
Date of publication: |
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03.08.2022 |
Supplementary Material: |
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InteractiveResource (Project Website): https://robertylewis.com/semilinear-paper/ |