Nonlinear Control Sy.. - Free

32 CHAPTER 2. MATHEMATICAL PRELIMINARIES
2.2 Metric Spaces
In real and complex analysis many results depend solely on the idea of distance between
numbers x and y. Metric spaces form a natural generalization of this concept. Throughout
the rest of the book, IR and C denote the field of real and complex numbers, respectively.
Z represents the set of integers. R+ and Z+ represent the subsets of non negative elements
of IR and Z, respectively. Finally Ilt"" denotes the set of real matrices with m rows and n
columns.
Definition 2.1 A metric space is a pair (X, d) of a non empty set X and a metric or
distance function d : X x X -4 ]R such that, for all x, y, z c X the following conditions hold:
(i) d(x, y) = 0 if and only if x = y.
(ii) d(x, y) = d(y, x).
(iii) d(x, z) < d(x, y) + d(y, z).
Defining property (iii) is called the triangle inequality. Notice that, letting x = z in
(iii) and taking account of (i) and (ii) we have, d(x, x) = 0 < d(x, y) + d(y, x) = 2d(x, y)
from which it follows that d(x, y) > 0 Vx, y E X.
2.3 Vector Spaces
So far we have been dealing with metric spaces where the emphasis was placed on the notion
of distance. The next step consists of providing the space with a proper algebraic structure.
If we define addition of elements of the space and also multiplication of elements of the
space by real or complex numbers, we arrive at the notion of vector space. Alternative
names for vector spaces are linear spaces and linear vector spaces.
In the following definition F denotes a field of scalars that can be either the real or
complex number system, IR, and C.
Definition 2.2 A vector space over F is a non empty set X with a function "+ ": X x X ->
X, and a function ". ": F x X --* X such that, for all A, p E F and x, y, z E X we have
(1) x + y = y + x (addition is commutative).
(2) x + (y + z) = (x + y) + z (addition is associative).