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

Abstract

We develop a theory of Hilbert $\widetilde{\C}$-modules by investigating their structural and functional analytic properties. Particular attention is given to finitely generated submodules, projection operators, representation theorems for $\widetilde{\C}$-linear functionals and $\widetilde{\C}$-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.

References in corpus (2)

• Internal sets and internal functions in Colombeau theory
• Ideals in the ring of Colombeau generalized numbers

Cited by in corpus (1)

• Foundations of the calculus of variations in generalized function algebras