I - Elements of functional analysis

Abstract

This chapter discusses the fundamental results of functional analysis. It firstly presents the notion of a Hilbert space and discusses some basic properties of the orthogonal projection operator. It then introduces the concept of closeness and completeness of a system of elements that belong to a Hilbert space. The completeness of the system of elementary sources is a necessary condition for the solution of scattering problems in the framework of the discrete sources method. After this discussion, the chapter briefly presents the notions of Schauder and Riesz bases. This concept will be used to analyze the convergence of the null-field method. A convergent projection scheme is constructed by appealing on the fundamental theorem of discrete approximation. The chapter concludes by analyzing projection methods for a linear operator “A” acting from a Hilbert space “H” onto a Hilbert space G, and for the operator equation.