Outline of a Theory of Functions of an Abstract Variable ar 
Outline of a Theory of Functions of an 
Abstract Variable 
The object of this paper is simply to develop, in outline, a 
theory of functions, analytic in the same sense as that used in 
the theory of the complex variable, but for a variable whose 
range may have as wide a meaning as possible. In the problem 
of integration, however, we need not be restricted to analytic 
functions. More precisely formulated, the aim is to develop 
mathematical operations so that very general types of equations 
of identity F(x) =0 which may occur in ordinary analysis are ° 
both significant and true in those domains of analysis for which 
the operations are defined. These domains occur as common 
illustrations of the Banach space. 
he linear vector space. This comprises the ordinary linear 
abstract space involving elements, called vectors or points, and 
three functional operations: addition, scalar multiplication, and 
forming the norm. We shall write the well-known governing 
postulates, partly for completeness, and partly to allow a perti- 
nent comment. 
Let E represent the space; x, y, 3, vectors in £; and &, 4, com- 
plex numbers, it being assumed that the associated number sys- 
tem, denoted by C, is complex. We shall use the ordinary 
symbols of arithmetic for addition and scalar multiplication. 
P.1. x+y is a function on E to E, called addition. 
P.2. xty=y+x 
P.3. 0 (x+y) +2=x+(y+z) 
P.4. £-x isa function on C and E to E, called scalar multi- 
plication. 
Gere E-(x+y)=x-E+E-y 
P28. (E+n)-x=E-x+n-x 
P98: (E-9)-x=£-(n-x) | 
P.10. ||x|| is a function on E to the positive real number 
