Page 280 - Matrix Analysis & Applied Linear Algebra

276 Chapter 5 Norms, Inner Products, and Orthogonality
Example 5.1.3

Equivalent Norms. Vector norms are basic tools for defining and analyzing
limiting behavior in vector spaces V. A sequence {x k }⊂V is said to converge
to x (write x k → x )if x k − x → 0. This depends on the choice of the norm,
so, ostensibly, we might have x k → x with one norm but not with another.
Fortunately, this is impossible in finite-dimensional spaces because all norms are
equivalent in the following sense.
Problem: For each pair of norms, , , on an n-dimensional space V,
a b
exhibit positive constants α and β (depending only on the norms) such that

x a
α ≤ ≤ β for all nonzero vectors in V. (5.1.10)
x
b
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y > 0, and write
b a
Solution: For S b = {y | y =1}, let µ = min y∈S b

x
x
∈S b =⇒ x = x

≥ x min y = x µ.
a
b
a
b
x b
x
y∈S b
b b a
The same argument shows there is a ν> 0 such that x ≥ ν x , so
b a
(5.1.10) is produced with α = µ and β =1/ν. Note that (5.1.10) insures that
x k − x → 0if and only if x k − x → 0. Specific values for α and β are
a b
given in Exercises 5.1.8 and 5.12.3.
Exercises for section 5.1
   
2 1+i
1 1 − i
5.1.1. Find the 1-, 2-, and ∞-norms of x =   and x =   .
−4 1
−2 4i
   
2 1
1 −1
5.1.2. Consider the euclidean norm with u =   and v =   .
−4 1
−2 −1
(a) Determine the distance between u and v.
(b) Verify that the triangle inequality holds for u and v.
(c) Verify that the CBS inequality holds for u and v.
2 2 2 2
5.1.3. Show that (α 1 + α 2 + ··· + α n ) ≤ n α + α + ··· + α for α i ∈ .
1 2 n
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An important theorem from analysis states that a continuous function mapping a closed and
bounded subset K⊂ V into attains a minimum and maximum value at points in K.
Unit spheres in finite-dimensional spaces are closed and bounded, and every norm on V is
continuous (Exercise 5.1.7), so this minimum is guaranteed to exist.