Outline of a Theory of Functions of an Abstract Variable 61 
I} mal] S |laey|l -[] al], 
which holds for all x and every Gn), there exists some value of x 
not zero for which the equality is true, we shall say that that 
norm is the most effective norm. In this case we have the theo- 
rem that if the norms of all the elements of E,,), for all m, are the 
most effective, the only nullifying element is the zero vector. 
When the norm is the most effective, two vectors which differ 
by a nullifying element are identical, otherwise they are merely 
equivalent with respect to multiplication. 
Multilinear functions. Let x1, %2,--~-, Xn denote a set of 
independent variables whose range is LE.) (note that the sub- 
scripts above are not in parentheses). 
We shall say that a function f(x) on Ea) to Ey) is homo- 
geneous of the mth degree if for all £ and x; 
fléa1) = E"f(x1); 
and linear homogeneous if 
fix) = Ef( xy) 
f(a + %2) = fla) + f' x2) 
for all x, and xp. 
It is readily seen that a,n)x1 is a linear homogeneous function 
which is also uniformly continuous. Since the product is an 
element of E,,_1) for each value of x;, we may multiply it on the 
right by x: of Eq, and so obtain the function a,,)x%1%2 on Eqf to 
Enz). This function of two independent variables is linear 
homogeneous in each, and it is therefore called bilinear. It is, 
moreover, uniformly continuous in both variables if their range 
is a bounded region. 
In a similar manner it is possible to consider the multilinear 
function with independent variables which is generated by 
Successive multiplication on the right: 
n a} 
@in)X¥i¥0%3 >: % On Ey, tO Bay, 
but we shall be chiefly interested in this function only when 
%1=%=---=y,. It is then called a monomial of degree”. 
Symmetric elements. The element a,,),” 22, is called a sym- 
metric element if the function 
AinyX1%2 °° Xn 
