In homogeneous coordinates, the first positive quadrant joins 
with 
by "points" 
, and is mapped onto the hyperbolic line 
by the correspondence 
. Now define
 |
(1)
|
Let 
be any bounded linear transformation of a Banach space 
that maps a closed convex
cone 
of 
onto itself. Then the 
-norm 
of 
is defined by
 |
(2)
|
for pairs 
with finite 
.
Birkhoff's inequality then states that if the transform 
of 
under 
has finite diameter 
under 
, then
 |
(3)
|
(Birkhoff 1957).
