84 MATHEMATICS: A. A. BENNETT Proc. N. A. S. 
NORMALIZED GEOMETRIC SYSTEMS 
By Albert A. Bennett 
University of Texas 
Communicated by E. H. Moore, January 18, 1921 
The notion of norm or numerical value of a complex quantity, c — a -f- 
feV— 1, namely, |c| = Va 2 + b 2 , as it arises in algebra, has a more or 
less immediate generalization to more extensive matric systems. The 
three important properties: "(1) \ci + c 2 \ < \ci\ + \c 2 \; (2) \ci.c 2 \ = [ei|.Jc 2 J; 
(3) c\ is the positive square root of a positive definite quadratic form, 
are carried over at the expense only of replacing (2) by (2') \ci.c 2 \ < \c1\.\c2l 
and allowing in place of (3), (3') \c\ is the positive square root of a positive 
definite form, Hermitian or quadratic. By C\.c 2 in these geometrical 
examples is meant the inner product 1 or a generalization of it. Two 
other generalizations of norm have been of great importance. The 
first of these is that of the theory of algebraic numbers, 2 where (1) is 
dropped, (2) is retained, and in place of (3) one has, Norm of c is a cer- 
tain function of the nth degree, n being the order of the algebraic field. 
The second is that of a general theory of sets as treated for example by 
Frechet, 3 where (1) is retained, (2) and (3) are dropped. The theory of 
integral equations as usually developed is geometrical in an infinity of 
dimensions and retains (1), (2'), (3'). It is noted that instances in which 
(1) and (2') are retained usually keep (3') also. Now the importance 
of (3') is chiefly that it implies (1) and (2') with the conventions as to 
linearity and so forth usually assumed. The converse that (1) and (2') 
imply (3') is false. It is of interest to show that most of the familiar 
properties of the norm may be retained, in particular (1) and (2'), when 
the norm is positive definite but otherwise largely arbitrary. 
Three discussions bearing on this topic may be referred to. First, 
a geometrical study involving points but not their duals, by Minkowski, 
in his Geometrie der Zahlen.* The great generality of the idea of norm is 
there beautifully developed although it is not carried so far as it is here; 
but since the concept of the point dual is not brought in by Minkowski, 
most of the ideas here discussed are not found there. A second discussion, 
involving inner products, but treating only a very special case of the 
non-quadratic norm for an infinite number of variables is given by F. 
Riesz 5 in examining the convergence of bilinear forms. The third dis- 
cussion involving only a scalar system, and hence without inner products, 
between elements of different systems is given by Kiirschak. 6 It is 
perhaps the most suggestive system of scalars in the literature in which 
\\n\\ may be less than n, for n a natural number. 
The following treatment relates under one head the notions of convex 
region, 4 the triangle property, 7 the linearly homogeneous property of 
distance 8 or norm, conjugate norms, 5 the inner product, 1 convergence of 
