TABLE OF CONTENTS For 
we denote by 
the set of all n-tuples x=(x 1 , x 2 , , x n ) of real numbers which, with respect to the operations + (internal composition law) and (external composition law) defined by
for every 
, and respectively by
for every 
and every 
, is an n-dimensional vector space over 
. In all that follows, 
is the standard inner product on , i.e.
and 
is the induced Euclidean norm, i.e.
for every 
. Whenever no confusion may occur, we will cancel the index n, writing 
x, y 
instead of x, y n and x instead of x n . Also, we will cancel by simply writing ?x instead of ? x.
Let 
be the set of all n m-matrices with real elements. In many situations we will identify an element 
by a linear operator (denoted for simplicity by the same symbol) 
, defined by
for every 
, where x is a column vector.
On the set 
, which clearly is an n m-dimensional vector space over , we define the function 
, by
for every 
. The next simple lemma is particularly useful in what follows.
The function 
is a norm on 
, i.e. it satisfies:
(N 1 ) 
if and only if 
is the null matrix;
(N 2 ) 
for every 
and every 
;
(N
