328 CHAPTER 13. NORMS
Corollary 13.0.8 Suppose X is a finite dimensional linear space with the field of scalarseither C or R and ||·|| and |||·||| are two norms on X. Then there exist positive constants, δand ∆, independent of x ∈ X such that
δ |||x||| ≤ ||x|| ≤ ∆ |||x||| .
Thus any two norms are equivalent.
This is very important because it shows that all questions of convergence can be consid-ered relative to any norm with the same outcome.
Proof: Let {v1, · · · ,vn} be a basis for X and let |·| be the norm taken with respect tothis basis which was described earlier. Then by Theorem 13.0.4, there are positive constantsδ1,∆1, δ2,∆2, all independent of x ∈X such that
δ2 |||x||| ≤ |x| ≤ ∆2 |||x||| , δ1 ||x|| ≤ |x| ≤ ∆1 ||x|| .
Then
δ2 |||x||| ≤ |x| ≤ ∆1 ||x|| ≤∆1
δ1|x| ≤ ∆1∆2
δ1|||x|||
and soδ2∆1
|||x||| ≤ ||x|| ≤ ∆2
δ1|||x||| ■
Definition 13.0.9 Let X and Y be normed linear spaces with norms ||·||X and ||·||Y re-spectively. Then L (X,Y ) denotes the space of linear transformations, called bounded lineartransformations, mapping X to Y which have the property that
||A|| ≡ sup {||Ax||Y : ||x||X ≤ 1} <∞.
Then ||A|| is referred to as the operator norm of the bounded linear transformation A.
It is an easy exercise to verify that ||·|| is a norm on L (X,Y ) and it is always the casethat
||Ax||Y ≤ ||A|| ||x||X .
Furthermore, you should verify that you can replace ≤ 1 with = 1 in the definition. Thus
||A|| ≡ sup {||Ax||Y : ||x||X = 1} .
Theorem 13.0.10 Let X and Y be finite dimensional normed linear spaces of dimensionn and m respectively and denote by ||·|| the norm on either X or Y . Then if A is any linearfunction mapping X to Y, then A ∈ L (X,Y ) and (L (X,Y ) , ||·||) is a complete normedlinear space of dimension nm with
||Ax|| ≤ ||A|| ||x|| .
Also if A ∈ L (X,Y ) and B ∈ L (Y, Z) where X,Y, Z are normed linear spaces,
∥BA∥ ≤ ∥B∥ ∥A∥
Proof: It is necessary to show the norm defined on linear transformations really is anorm. Again the first and third properties listed above for norms are obvious. It remains to