Outline of a Theory of Functions of an Abstract Variable 71 
Substituting this in (1) we obtain 
Sua =e Sn,0 _ O« 
Where every term ina is of the type 
n+1 Ss 
0 
Thus there are a finite number of terms of this type, s of them, 
infact. Each term approaches zero as n increases for the norm 
of (A) Smax,||x,"—"(xs41 ~x;)4||-ya0, where yas is the length of Tas, 
and its coefficient becomes vanishingly small. This proves 
(2). 
; 
Now on forming the sum pee it is seen that alternate 
terms cancel each other, so that 
- ; n+1 
dieSn.s= Fm) > (xt) — x7") = atm (b"t!—a"t), 
0 0 
and this is true for every value of 7. But by definition we have 
T=lim Sy. 
and an account of (2) 
r 
lim PRE He = (r+ 1) J. 
Hence J 
ee 2m) yet a), 
r+ 
which was to be proved. 
It is worthy of notice that the integral is independent of the 
path from a to b, so that if the uniform arc is closed, that is, 
a=b, the integral vanishes. 
Infinite series. Let {a} be an infinite sequence of elements 
all of which belong to some space Ejmj, and let us denote ~~ 
sum of the first ilies by S,, that is, 
Sp = a1 + a2 ae "+ On. 
Then the infinite series 
1 ap tt Gt 
is said to converge to the sum s if the sequence { be } converges 
to S. 
The series is said to converge absolutely if the corresponding 
series of positive numerical terms 
|Jaxl] + |lasl] + --- + lanl] + 
converges. 
There are two very useful theorems in connection with ab- 
Solute convergence; namely, 
