Preliminary Information about Integration Theory

Abstract

We use the following standard notations for some important: R is the set of all real numbers, N is the set of natural numbers, C is the set of all complex numbers, ∅ is the empty set. The symbol R ⁿ denotes the n-dimensional vector space of points x = (x ₁, x ₂ …, x _n), where x _i ∈ R, i = 1, 2, …, n. For arbitrary vectors x = (x ₁, x ₂ …, x _n), y =(y ₁ y ₂, …, y _n), we set

$$\left\langle {x,y} \right\rangle = {{x}_{1}}{{y}_{1}} + {{x}_{2}}{{y}_{2}} + \ldots + {{x}_{n}}{{y}_{n}}$$

The magnitude <x, y> is called the scalar product of the vectors x and y. For x ∈ R, |x| denotes the length of x, i. e., |x| = (<x, y>)^1/2.