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

TL;DR

This paper develops a comprehensive theory of Hilbert C-modules, exploring their structural properties and applying them to solve variational problems with generalized forms using a generalized Lax-Milgram theorem.

Contribution

It introduces new structural and functional analytic results for Hilbert C-modules and extends the Lax-Milgram theorem to this setting for variational problem solutions.

Findings

Characterization of finitely generated submodules

Representation theorems for C-linear functionals

Existence and uniqueness results for 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.