Chapter 5 - Preliminary Mathematical Concepts

Abstract

This chapter focuses on some preliminary mathematical concepts. A normed space is a vector space on which a real-valued transformation is defined such that it assigns to each element. The procedure by which an orthonormal basis is obtained is called the Gram–Schmidt procedure. The determinant of the Gram matrix is referred to as the Gram determinant. In a normed space, every convergent sequence is a Cauchy sequence. If a sequence converges in a normed space, its limit is unique. A complete normed space is called a Banach space. A complete vector space X together with an inner product defined on X × X is called a Hilbert space. In a normed space, any finite-dimensional subspace is complete. Every subset of a countable set is either a finite set or a countable set, and the set of all finite sequences from a countable set is countable.