Introduction to the subject. 1.Variational problems, the Ritz method, and the idea of orthogonality. 2.The galerkin method for differential and integral equations, the friedrichs extension, and the idea of Self-Adjointness. 3.Difference methods and stability. Linear monotone problems. 4.Auxiliary tools and the convergence of the galerkin method for linear operator equations. 5.Hilbert space methods and linear elliptic differential equations 6.Hilbert space methods and linear parabolic differential equations. 7.Hilbert space methods and linear hyperbolic differential equations.