Banach and Hilbert Spaces – Fourier Series and Orthogonal Polynomials
Gérard Favier · 2019
This chapter introduces the basic concepts underlying Banach and Hilbert spaces and gives an illustration thereof through the problem of function approximation in the form of expansions over Hilbert bases. Two types of bases are considered: trigonometric bases leading to Fourier series and bases of orthogonal polynomials. The chapter first defines the notions of distance and metric space. Then, it illustrates the use of distance for the study of convergence of sequences and of local continuity of a function. The notions of orthogonality, orthonormal basis, and orthogonal projection onto a subspace are presented before describing the Gram–Schmidt orthonormalization process which is a fundamental method of linear algebra. The chapter also presents the notion of Fourier series expansion of periodic functions. It makes the link between Hilbert spaces and function approximation through the use of expansions over Hilbert bases.