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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hilbert $ \widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems


Authors: Claudia Garetto and Hans Vernaeve
Journal: Trans. Amer. Math. Soc. 363 (2011), 2047-2090
MSC (2000): Primary 46F30, 13J99
Published electronically: November 2, 2010
MathSciNet review: 2746675
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Abstract: We develop a theory of Hilbert $ \widetilde{\mathbb{C}}$-modules which forms the core of a new functional analytic approach to algebras of generalized functions. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $ \widetilde{\mathbb{C}}$-linear functionals and $ \widetilde{\mathbb{C}}$-sesquilinear forms. We establish a generalized Lax-Milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.


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Additional Information

Claudia Garetto
Affiliation: Institut für Grundlagen der Bauingenieurwissenschaften, Arbeitsbereich für Technische Mathematik, Leopold-Franzens-Universität, Innsbruck, Austria
Address at time of publication: Department of Mathematics, Imperial College London, London, United Kingdom
Email: claudia@mat1.uibk.ac.at, c.garetto@imperial.ac.uk

Hans Vernaeve
Affiliation: Institut für Grundlagen der Bauingenieurwissenschaften, Arbeitsbereich für Technische Mathematik, Leopold-Franzens-Universität, Innsbruck, Austria
Address at time of publication: Department of Mathematics, Ghent University, Ghent, Belgium
Email: hans.vernaeve@uibk.ac.at, hvernaeve@cage.UGent.be

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05143-8
PII: S 0002-9947(2010)05143-8
Received by editor(s): November 15, 2007
Received by editor(s) in revised form: June 22, 2009
Published electronically: November 2, 2010
Additional Notes: The first author was supported by FWF (Austria), grants T305-N13 and Y237-N13
The second author was supported by FWF (Austria), grants M949-N18 and Y237-N13.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.