86 
MATHEMATICS: A. A. BENNETT 
Proc. N. A. S. 
The "Regular Multiplication Proposition," R. Mult. (R, S, T, P): 
(i) Reg. (R), Reg. (S), Reg. (T), Reg. (P). (ii) Mult. {R, S, T, P). 
(Hi) Mult. (i? c *, S 0 *, T 0 *, P G *). (w) Mult. (R ot S Q , T 0 , P Q ). 
The "Regular Commutative Proposition," R. Com. (C, M, L): (i) Com. 
(C, M, L). (ii) For every c of C and I of L, c 0 .l 0 , c 0 .l* and c*.l 0 are of 
L D ; c* ./* is of L* ; c ./* , c* ./ oo » and c oo ^ oo are of £ oo • lc * s of L* . 
(iv) For every ^ and h of L, if neither h nor / 2 is in L qq , /i -f- Z 2 is not in 
L qq , and if both Zi and Z 2 are in L Q , Zi + Z 2 is in L Q . 
The "Regular Conjugate Proposition," R. Con]. (C, i?, S) : For each 
r of R, there is a c of C and an 5 of 5, such that (i) r.s = s.r = c.c; (ii) 
if r is of R Q , c is of C 0 , 5 is of S Q ; if r is oi R*, c is of C*, 5 is of S* ; if r 
is of oq , c is of C , 5 is of 5 ^ . 
The "Normal Proposition," iVVm. (L) : (*) Each I of L has associated 
with it a unique value, |(/||. (ii) Reg. (L). (m) For each l Q , \\l 0 \\ — 0; 
for each /*, |(Z*|| is a positive real finite number; for each I ^ , [|^ 00. f| * s 
positively infinite, (iv) For each I of L, || — Z|( = ||/||. (v) If l L exists, 
\\l h \\ = 1. (vi) There exists at least one Z* for which ||Z*|| is different from 
unity. 
The "Normal Multiplication Proposition," N. Mult. (R, S, T, P): 
(i) Nrm. (R), Nrm. (5), Nrm. (T), Nrm. (P). (ii) Mult. (R, S, T, P). 
da) \\s.t\\ < H4II4 
The "Normal Commutative Proposition," N. Com. (C, M, L): (i) 
Com. (C, M y L). (ii) For every c of C and Z of L, for which ||c||.[|Z|| exists, 
\\c.l\\ =■ ||4II% (w) ||lc|i = 1. («0 For every h and Z 2 of L, + bll 
< \U + IM. 
The "Normal Conjugate Proposition," N. Con]. (C } R, S): For each 
r of R, there is a c of C and an s of 5, such that (i) r.s = s.r = c.c, (ii) 
INI = [Ml = 114 
The "Linear Proposition," liw. (C, M, L) : (i) Mult. (M, M, L, L). 
(«) Mult. (M, L, M, L). (Hi) Com. (C, M, L). 
In the above, L is called a linear system, with M as system of multipliers, 
and C as commutative subsystem of M. 
The "Hypernumber Proposition," Hyp. (C, H): (i) Lin. (C, C, C). 
(ii) Lin. (C,H,H). 
In the above, H is called a system of hypernumber s with C as commutative 
subsystem. 
The "Vector Proposition," Vect. (C, H, V): (i) Hyp. (C, H). (ii) 
Lin. (C, H, V). 
In the above, V is called a system of vectors, with H as associated system 
of hypernumbers. 
The "Regular Linear Proposition," R. Lin. (C, M, L) : 0') R. Mult. 
(Af, M, L, L). (ii) R. Mult. (M, L, M, L). (Hi) R. Com. (C, M, L). 
The "Regular Hypernumber Proposition," R. Hyp. (C, H) : (i) R. Lin. 
(C, C, C). (ii) R. Lin. (C, H, H). 
