HILBERT (C)over-tilde-MODULES: STRUCTURAL PROPERTIES AND APPLICATIONS TO VARIATIONAL PROBLEMS

We develop a theory of Hilbert (C) over tilde -modules which forms the core of a new functional analytic approach to algebras of generalized functions. Particular attention is given to finitely generated subrnodules, projection operators, representation theorems for (C) over tilde -linear functionals and (C) over tilde -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.