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2.5 Vector Spaces, Norms, and Inner Products 41

3. ||x+y||≤||x||+|y|| for all x,y ∈X,

EXAMPLE 2.5–2: Vector Norms (1)

n
The following are common norms in X= where is the set of n×1 vectors
n
with real components.
1. 1-norm:
2. 2-norm: also known as the Euclidean norm

3. p-norm:
4. 8-norm:

EXAMPLE 2.5–3: Vector Norms (2)
Consider the vector

Then, ||x|| 1=5, ||x|| 2=2 and ||x||∞=2.

n
We now present an important property of norms of vectors in which will
be useful in the sequel.

LEMMA 2.5–1: Let ||x|| a and ||x|| b be any two norms of a vector x∈ . Then
there exists finite positive constants k 1 and k 2 such that

The two norms in the lemma are said to be equivalent and this particular

property will hold for any two norms on .

Copyright © 2004 by Marcel Dekker, Inc.