Hilbert Spaces

Abstract

A Hilbert space is an abstract vector space with the following two properties: the inner product property (Sect. 4.1.3), which determines the geometry of the vector space, and the completeness property (Sect. 4.1.6), which guarantees the self-consistency of the space. Most of the mathematical topics covered in this volume are AQ1 based on Hilbert spaces. In particular, L^p spaces and l^p spaces (Sect. 4.3), which are specific classes of Hilbert spaces, are crucial for the formulation of the theories of orthonormal polynomials, Lebesgue integrals, Fourier analyses, and others, as we discuss in subsequent chapters.