Continuity and Galerkin Method

This lecture covers the concept of continuity in function spaces, focusing on the continuity of linear operators. It introduces the Galerkin method for solving boundary value problems, emphasizing the importance of well-posedness and uniqueness. The instructor explains the application of the Galerkin method in finding weak solutions and the use of Hilbert spaces. The lecture also discusses the construction of finite element spaces and grids for numerical methods, highlighting the difference between Galerkin method and finite difference method. Various inequalities and error estimations are presented, showcasing the optimality of the Galerkin method in approximating solutions.