Hilbert ℂ̃-modules: structural properties and applications to variational problems

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.