Hilbert ̃ℂ-modules: Structural properties and applications to variational problems

Garetto, Claudia; Vernaeve, Hans

doi:10.1090/S0002-9947-2010-05143-8

Hilbert $\widetilde {\mathbb {C}}$-modules: Structural properties and applications to variational problems
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by Claudia Garetto and Hans Vernaeve PDF

Trans. Amer. Math. Soc. 363 (2011), 2047-2090 Request permission

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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