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2.5 Vector Spaces, Norms, and Inner Products 45

2. which measures the

total energy.

EXAMPLE 2.5–6: Function Norms

-t
1. The function f(t)=e belongs to L 1. In fact, ||e || 1=1. The function
-t
belongs to L 2. The sinusoid f(t)=2sint belongs to L ∞ since its
magnitude is bounded by 2 and ||2sint|| ∞= 2.
2. Suppose the vector function x(t) has continuous and real-valued
components, i.e.

where [a, b] is a closed-interval on the real line R. We denote the set of such
functions x by n [a, b]. Then, let us define the real-valued function

where ||x(t)|| is any previously defined norm of x(t) for a fixed t. It can be
verified that ||x(.)|| is a norm on the set n [a, b] and may be used to compare
the size of such functions [Desoer and Vidyasagar 1975]. In fact, it is very
important to distinguish between ||x(t)|| and ||x(.)||. The first is the norm of a
fixed vector for a particular time t while the second is the norm of a time-
dependent vector. It is this second norm (which was introduced in definition
2.5.6) that we shall use when studying the stability of systems.

-2 T
-t
-t
3. The vector f(t)=[e -e -(1+t) ] is a member of . On the other hand,
-1 T
f(t)=[e -e -(1+t) ] is a member of and .
-t
-t

In some cases, we would like to deal with signals that are bounded for finite
times but may become unbounded as time goes to infinity. This leads us to
define the extended L p spaces. Thus consider the function
(2.5.1)

then, the extended L p space is defined by

Copyright © 2004 by Marcel Dekker, Inc.