Outline of a Theory of Functions of an Abstract Variable 59 
lesa ee 
The elementary theorems which show the existence of a zero 
element, uniqueness of the limit, unique solvability of linear 
equations, etc., are well-known and we shall not bother to state 
them. 
The generalized vector space. We consider the following pos- 
tulates as a basis for the ensuing analysis. 
. For every positive integer m and including 0 there exists 
a complete linear vector space E(x), containing elements x(,). 
II. There exists an operation called vector multiplication, 
which has the following properties. 
P.15. For m40, n#0, m=n, XcmyVny iS a function on E ym) 
900 Foo tO Bigg ncn. 
P.16. (E+ Xm) (9° ¥ ny) = (E- 9) - (Xm) Yiny)- 
P17. 26) (Y (ny +E (ny) = (2 ¢m)Veny) + (H emyXtn)) 
P.18. (% (my) + (my) Biny = (% (m2 (ny) + (Y (m2 ny) + 
E19. Xin) (Vimy Sep) = (Lom Veny) Bip), WF 1. 
P.20. The equation X(m)y@(ny = D(mtn—2), (ny #0, has at least 
one solution for xm). 
P.21. | 2 cmy¥ cny| S||2¢m)l| -|[yemll- 
> *pVa)~ VayXq)- 
A system having the properties stated in I and II is called 
a generalized vector space. 
Notation. Evidently we have not overtly distinguished be- 
tween various types of addition and multiplication. The exact 
operation will always be clear from the nature of the elements 
involved; as well as the meaning of the symbol 0, which will 
also acquire a unique significance in its context. For conven- 
ience, elements x,) of Eq) will be more simply denoted by x, 
leaving the letter free for other distinguishing subscripts. 
Continuity. A function f(x) on S(S¢ E,) to S’(S’ ¢€ E,y) 1s 
said to be continuous on S if for every 7 >0 there exists a 6 de- 
pending, in general, upon 7 and x, such that lly —2| <6, where 
yeS, implies that f(y) —f(x)|| <n. If dis independent of x, then 
f(x) is said to be uniformly continuous throughout S. 
The M-function. A function f(x) on S to S’, as above, is 
called an M-function if there exists a sequence of sets of points 
in E.;, denoted by {S,}, where S,¢S,i1 and S is the outer 
limiting set of the sequence, such that f(x) is continuous oS Sn 
for every n. It is not difficult to see that if S is a region in ‘es 
space E which possesses a Lesbegue measure and each S, is_ 
