88 
MATHEMATICS; A. A. BENNETT 
Proc. N. A. S- 
To normalize a geometric system admitting a guage set, define for 
every x G and u G , \\x G \\ = 1, \\u G \\ = 1, and for every c.x G and c.u G , 
\\c-%g\\ — lk|L \\ c - u g\\ = IM1- The theorem then follows without diffi- 
culty. 
For the general normal geometric system, the relative inclination, 6, 
of two elements, x of X* , and u of U*, may be defined by, \\x.u\\ = 
cos 0. Thus x of X*, and w of L 7 * are mutually orthogonal if and only 
if = 0. Two elements, # of X* and w of £/*, are mutually conju- 
gate if and only if x.u = u.x = c.c while also \\x\\ = = Two 
elements, x of X*, and u of [/*, are mutually reciprocal if and only if = 
w.z = l c while also ||*||.[|«[| = 1. The sets (x) and (u) for which 
= 1, and ||w|| = 1, respectively, are Convex. The relation ||fi + 
rj| < + IMI is called the Triangle Inequality of the norm. The 
relation = is called the Linearly Homogeneous Property 
of »the norm. The expressions \\x\\ and \\u\\ are said to be Conjugate 
Norms and when continuous, each is a "Gauge Form." The expressions 
x.u and u.x are Inner Products. Each of the bilinear forms x.u and u.x 
"converges" if \\x\\ and are finite, since \\x.u\\ < ~and 
\\u.x^ < m|.||^||, which are statements of the general form of Schwarz's 
Inequality. 
The following possibilities for a normal geometrical system may be 
emphasized: (i) The product among hypernumbers need not be com- 
mutative, (it) The product involving two #*s or two u's need not have 
a meaning, (tit) The system C may be an integral system in which divi- 
sion is not in general possible, (iv) \\nc\\ may be less than n, where by 
n c is meant l c + l c + • • • + lc to n terms, for n > 1. (v) The conju- 
gate of a given element need not be unique, (vi) The set of elements 
(x) for which = 1, may be dense but not continuous, as for example 
the rational points on a circle. In particular, the above theory is ap- 
plicable to the Geometry of Numbers of Minkowski, 4 to a system where 
C is a Kiirschak valuated field, 6 and to a system where H is a quaternion 
field. 12 
While in every normal geometric system each x of X* has a conjugate, 
u of £/*, which has again the given x as its conjugate, the conjugate 
relation is not in general a simple one. The geometric systems for which 
(3') is satisfied are identified by the semi-linear property that if %\ and Ui 
be conjugates, and x 2y u 2 be conjugates, then Ci.xi + c 2 x 2 and c\.Ui + c' 2 .u 2 
will be conjugates where Ci, c 2 , c\, c' 2 are of C, while c and c' are themselves 
conjugates in a more elementary sense. 
1 Due to Grassmann, Geometrische Analyse, 1847, paragraph 7, p. 16; cf. Gibbs, 
Vector Analysis (E. B. Wilson). 
2 Suggested by Gauss, Dirichlet, Werke, I, p. 539. Extended use by Kronecker, 
"De unitatibus complexis," Werke I, p. 14. 
3 Frechet, M., Sur quelques points..., Rend. Circ. M. Palermo, 22, 1906 (1-74). 
