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40 Introduction to Control Theory

8. For each x∈V, we have 1.x=x where 1 is the unity in (resp. in ).

EXAMPLE 2.5–1: Vector Spaces
The following are linear vector spaces with the associated scalar fields:
with , and n with .

DEFINITION 2.5–2 A subset M of a vector space V is a subspace if it is a
linear vector space in its own right. One necessary condition for M to be a
subspace is that it contains the zero vector.
We can equip a vector space with many functions. One of which is the
inner product which takes two vectors in V to a scalar either in or in the
other one is the norm of a vector which takes a vector in V to a positive
value in . The following section discusses the norms of vectors which is
then followed by a section on inner products.

Norms of Signals and Systems
A norm is a generalization of the ideas of distance and length. As stability
theory is usually concerned with the size of some vectors and matrices, we
give here a brief description of some norms that will be used in this book. We
will consider first the norms of vectors defined on a vector space X with the
associated scalar field of real numbers then introduce the matrix induced
norms, the function norms and finally the system-induced norms or operator
gains.

Vector Norms

We start our discussion of norms by reviewing the most familiar normed
spaces, that is the spaces of vectors with constant entries. In the following,
||a|| denotes the absolute value of a for a real a or the magnitude of a if a is
complex.

DEFINITION 2.5–3 A norm ||·|| of a vector x is a real-valued function defined
on the vector space X such that

1. ||x||>0 for all x∈X, with ||x||=0 if and only if x=0.
2. ||αx||=|α|||x|| for all x∈X and any scalar α.

Copyright © 2004 by Marcel Dekker, Inc.