Nonlinear Control Sy.. - Free

38 CHAPTER 2. MATHEMATICAL PRELIMINARIES
Example 2.7 Consider again the vector space IRn. For each p, 1 < p < oo, the function
lip, known as the p-norm in Rn, makes this space a normed vector space, where
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In particular
IlxiiP f (Ix1IP + ... + IxnIP)1/P
Ilxlll
IIxlI2
def
def
Ix11
+... Ixnl. (2.6)
I21Iz+...+Ixnl2. (2.7)
The 2-norm is the so-called Euclidean norm. Also, the oo-norm is defined as follows:
IIxiI. dcf maxlx,I. (2.8)
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By far the most commonly used of the p-norms in IRn is the 2-norm. Many of the theorems
encountered throughout the book, as well as some of the properties of functions and
sequences (such as continuity and convergence) depend only on the three defining properties
of a norm, and not on the specific norm adopted. In these cases, to simplify notation, it
is customary to drop the subscript p to indicate that the norm can be any p-norm. The
distinction is somewhat superfluous in that all p-norms in 1R are equivalent in the sense
that given any two norms II - IIa and II - IIb on ]Rn, there exist constants k1 and k2 such that
(see exercise (2.6))
ki llxlla 1. Then
Ilyllq , dx, y E IRn. (2.9)
llx + yIIP IIx1IP + IIyIIP , Vx, y E IRn. (2.10)