Linear Algebra, Theory And Applications, 2012a

342 NORMS FOR FINITE DIMENSIONAL VECTOR SPACES
Here is another proof of this ( proposition. ) Recall there are unitary matrices of the right
σ 0
size U, V such that A = U
V
0 0
∗ where the matrix on the inside is as described
in the section on the singular value decomposition. Then since unitary matrices preserve
norms,
( ||A|| = sup
σ 0
∣ U |x|≤1
0 0
( = sup
σ 0
∣ U 0 0
|y|≤1
)
( )
V ∗ x
∣ = sup
σ 0
∣ U V ∗ x
|V ∗ x|≤1
0 0 ∣
)
( )
y
∣ = sup
σ 0
∣ y
0 0 ∣ = σ 1 ≡||A|| 2
|y|≤1
This completes the alternate proof.
From now on, ||A|| 2
will mean either the operator norm of A taken with respect to the
usual Euclidean norm or the largest singular value of A, whichever is most convenient.
An interesting application of the notion of equivalent norms on R n is the process of
giving a norm on a finite Cartesian product of normed linear spaces.
Definition 14.0.14 Let X i ,i=1, ··· ,n be normed linear spaces with norms, ||·|| i
. For
define θ : ∏ n
i=1 X i → R n by
x ≡ (x 1 , ··· ,x n ) ∈
n∏
i=1
X i
θ (x) ≡ (||x 1 || 1
, ··· , ||x n || n
)
Then if ||·|| is any norm on R n , define a norm on ∏ n
i=1 X i, also denoted by ||·|| by
||x|| ≡ ||θx|| .
The following theorem follows immediately from Corollary 14.0.8.
Theorem 14.0.15 Let X i and ||·|| i
be given in the above definition and consider the norms
on ∏ n
∏ i=1 X i described there in terms of norms on R n . Then any two of these norms on
n
i=1 X i obtained in this way are equivalent.
or
For example, define
||x|| 1
≡
n∑
|x i | ,
i=1
||x|| ∞
≡ max {|x i | ,i=1, ··· ,n} ,
( n
) 1/2
∑
||x|| 2
= |x i | 2
i=1
and all three are equivalent norms on ∏ n
i=1 X i.