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2.5 Vector Spaces, Norms, and Inner Products 43

EXAMPLE 2.5–5: Induced Matrix Norms

Consider the ∞ induced matrix norm, the 1 induced matrix norm and the 2
induced matrix norm,

where max is the maximum eigenvalue. As an illustration, consider the matrix

Then, ||A|| i1=max(4, 4, 5)=5, ||A|| i2=4.4576, and ||A|| i∞= max(4, 7, 2)=7.

Function Norms

Next, we review the norms of time-dependent functions and vectors of
functions. These constitute an important class of signals which will be
encountered in controlling robots.

DEFINITION 2.5–5 Let f(.): [0, ∞)→R be a uniformly continuous function.
A function f is uniformly continuous if for any >0, there is a ( ) such that

Then, f is said to belong to L p if for p∈[1, ∞),

f is said to belong to L ∞ if it is bounded i.e. if
where sup f(t) denotes the supremum of f9t) i.e. the smallest number that is

Copyright © 2004 by Marcel Dekker, Inc.