Nonlinear Control Sy.. - Free

46 CHAPTER 2. MATHEMATICAL PRELIMINARIES
Example 2.10 Let X = (0,1) (i.e., X = {x E R : 0 < x < 1}), and let d(x,y) =I x - y 1.
Consider the sequence {xn} = 1/n. We have
d(xn,xm.) =I 1- 1 I< 1+ 1 < 2
n m n m N
where N = min(n,m). It follows that {xn} is a Cauchy sequence since d(xn, xm) < e,
provided that n, m > 2/e. It is not, however, convergent in X since limn, 1/n = 0, and
An important class of metric spaces are the so-called complete metric spaces.
Definition 2.14 A metric space (X, d) is called complete if and only if every Cauchy sequence
converges (to a point of X). In other words, (X, d) is a complete metric space if for
every sequence {xn} satisfying d(xn,xm) < e for n, m > N there exists x E X such that
d(xn, x) -> 0 as n -> oo.
If a space is incomplete, then it has "holes". In other words, a sequence might be "trying" to
converge to a point that does not belong to the space and thus not converging. In incomplete
spaces, in general, one needs to "guess" the limit of a sequence to prove convergence. If
a space is known to be complete, on the other hand, then to check the convergence of a
sequence to some point of the space, it is sufficient to check whether the sequence is Cauchy.
The simplest example of a complete metric space is the real-number system with the metric
d =1 x - y 1. We will encounter several other important examples in the sequel.
2.7 Functions
Definition 2.15 Let A and B be abstract sets. A function from A to B is a set f of ordered
pairs in the Cartesian product A x B with the property that if (a, b) and (a, c) are elements
of f, then b = c.
In other words, a function is a subset of the Cartesian product between A and B where
each argument can have one and only one image. Alternative names for functions used in
this book are map, mapping, operator, and transformation. The set of elements of A that
can occur as first members of elements in f is called the domain of f. The set of elements
of B that can occur as first members of elements of f is called the range of f. A function
f is called injective if f (xl) = f (X2) implies that xl = x2 for every x1, x2 E A. A function
f is called surjective if the range of f is the whole of B. It is called bijective if it is both
injective and surjective.