Investigation of the problem of representation of a linear functional in the form of a scoal product

Subject of research: the problem of representing a linear functional in the form of a scalar product in the Hilbert space. For example, this may be the Dirichlet problem for a second-order elliptic equation in variational form. Purpose of research: to present a general scheme of the method of iterative extensions for solving elliptic boundary value problems with Dirichlet conditions. Methods and objects of research: the object of study in such a problem can be the deformation of the membrane. A continuation of the considered problem in the Hilbert space is given. The extended problem is considered in a subspace of the Hilbert space. The extended problem is studied by the method of iterative extensions in the Euclidean space. An algorithm for implementing the method of iterative extensions is given. Main results of research: as a result, it turns out that the general scheme of the method of iterative extensions as applied to the solution of a boundary value problem with Dirichlet conditions for an elliptic equation does not depend on the order of this equation. An example is given to illustrate the conclusion about the efficiency of the method of iterative extensions.