Page 81 - Matrices theory and applications

4. Norms
64
4.1.1 Duality
n
n
Definition 4.1.2 Given a norm · on IR , its dual norm on IR
defined by
T
y x

.
x := sup
y =0 y
n is

The fact that · is a norm is obvious. The dual of a norm on CC is
n
T
defined in a similar way, with y x instead of y x. For every x, y ∈ CC ,
∗
one has
∗

y x ≤ x · y . (4.3)
Proposition 4.1.2 shows that the dual norm of · p is · q for 1/p+1/q =1.
This suggests the following property.
Proposition 4.1.4 The bidual (dual of the dual norm) of a norm is this
norm itself:

( · ) = · .
Proof

From (4.3), one has ( · ) ≤ · . The converse is a consequence of
the Hahn–Banach theorem: the unit ball B of · is convex and compact.
If x is a point of its boundary (that is, x = 1), there exists an IR-
affine (that is, of the form constant plus IR-linear) function that is zero
at x and nonpositive on B. Such a function can be written in the form
z → z y + c,where c is a constant, necessarily equal to − z x. Without
∗
∗
loss of generality, one may assume that z x is real. Hence
∗
∗
∗

y =sup y z = y x.
z =1
One deduces
y x
∗

( x ) ≥ =1 = x .
y
n
By homogeneity, this is true for every x ∈ CC .
4.1.2 Matrix Norms
Let us recall that M n (K) can be identified with the set of endomorphisms
n
of E = K by
A → (x → Ax).
Definition 4.1.3 If · is a norm on E and if A ∈ M n (K), we define
Ax
A := sup .
x =0 x