GARETTO, C. and VERNAEVE, H., 2011. Hilbert ℂ̃-modules: structural properties and applications to variational problems. Transactions of the American Mathematical Society, 363 (4), pp. 2047-2090.
We develop a theory of Hilbert ℂ̃-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 ℂ̃-linear functionals and ℂ̃-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.
First published in Transactions of the American Mathematical Society in 2011, published by the American Mathematical Society