Hilbert $\tilde{\text{C}}$-modules: structural properties and applications to variational problems

We develop a theory of Hilbert $\tilde{\text{C}}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\tilde{\text{C}}$-linear functionals and $\tilde{\text{C}}$-sesquilinear forms. By making use of a generalized Lax-Milgram theorem, we provide some existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form.