Outline of a Theory of Functions of an Abstract Variable 73 
T.28. If pz is the radius of convergence in the # complex- 
plane for a given power series, and if p is the greatest lower 
bound of p- as & varies over all unit elements in Ex), then the 
series converges uniformly and absolutely for all values of x for 
which ||x|| <p. 
The positive number p, which may also be 0 or ~, is called 
the radius of uniform convergence. The name is justified by 
the following theorem, whose proof is obvious. 
T.29. Let p be the radius of uniform convergence of a given 
power series, and a@ a positive number such that a>p. Then 
there exists some x such that || x]| =qa and for which the series 
diverges. 
The aggregate of values for which a series converges uni- 
formly may be referred to as the sphere of convergence. 
Regular functions. A function f(x) is said to be regular at 
the point x = xp if it is equal to a power series pC irtn} (x —Xo)” 
everywhere within the interior of a sphere whose center is at Xo 
and whose radius p is greater than zero. In this p neighborhood 
of xo, s.ch a function possesses certain regular properties, some 
of which are Ccisclosed in the following theorems. The proofs 
of these theorems are, as we expect, identical in form with their 
analogues in the theory of the complex variable; therefore, as 
usual, we omit them. 
T.31. If f(x) is regular at the point x =x» it is regular at 
every point interior to the sphere of webb bona of its repre- 
sentative power series. 
T.32. Let f(x) be regular at x =x, and let p be the radius of 
uniform convergence of the representative power series. Also 
let {x;} be an infinite sequence of points such that whatever be 
n and the unit element # there is always a term x;, i>m, such 
that x; - £% for some £, and such that the sequence has x» for a 
limit point. If f(x) =0 on every point of the sequence {xi} 
then f(x) =0 throughout the sphere of convergence. (It may 
easily be seen that in order to carry the proof through in the 
same fashion as the corresponding theorem in ordinary analysis, 
it is necessary to strengthen the hypothesis as we have done). 
Corollary I. Let f(x) and g(x) be regular in the neighbor- 
hood of a point x=x» and equal over such an infinite set of 
Points as described above. Then f(x) =g(x) throughout the 
neighborhood. 
Corollary II. If f(x) is regular at the point x ~ xo, and there 
exist two power series such that in a neighborhood of x». 
