Fundamentals

We begin by introducing here the mathematical notations and the basic results that will be used throughout this book. A (real) Vector spacevector space is a set X, whose elements are called Vectorsvectors, and in which two operations, addition and scalar multiplication, are defined as follows: (i) To every pair of vectors x and y corresponds a vector \(x + y\) in such a way that \(x + y = y + x\ \ \hbox {and} \ \ x + (y + z) = (x + y) + z.\) X contains a unique vector 0 (the zero vector or origin of X) such that \(x+0 = x\) for every \(x\in X\) , and to each \(x\in X\) corresponds a unique vector \(-x\) such that \(x + (-x) = 0.\) .