310 
MATHEMATICS: W. L. HART 
Definition 1. A function /(^) defined in R is completely continuous 
at a point ^^^^ — in notation, Ci (^^^0 — whenever 
lim Xin = Xi''\ i = 1,2, . . (3) 
it follows that 
W = 00 
This concept of complete continuity is the same as that of 'Vollstetig- 
keit^ which has been much used by Hilbert and his followers. However, 
in the present paper the functions satisfying Definition (1) are supposed 
defined for a region of points different from that used by Hilbert and 
entirely different applications of the concept are made. 
A function / (^) is said to be Co at ^^^^ if it has the weaker continuity 
resulting from Definition (1) when in (3) it is assumed that the con- 
vergence is uniform with respect to i. 
In deriving theorems on completely continuous functions important 
use is made of the following condensation lemma :^ 
Lemma 1. Let S = {^n- n = 1, 2, . . .) he a sequence of points of 
R. Then there exists a point ^' of R and an infinite sub-sequence 
^' n {n = 1, 2, . . .) of S such that lim^^^x,^ = Xi (i = 1, 2, . . 
The theorems obtained, of which the more important ones are given 
below, are derived by methods similar to those used in obtaining cor- 
responding results in classical analysis. 
Theorem 1. To state that f (^) is Co at ^^^^ is equivalent to saying that, 
for every e > 0 there exists a number de > 0 such that for every ^ of 
R satisfying \ x[—x^^- \ ^ de, i = 1, 2, . . ., there is the inequality 
fin- 
On taking this equivalent definition of the Co property, there is 
obtained 
Theorem 2. // / (^) is Ci at every point of R, then f (^) possesses the 
Co property uniformly in R. 
The analogues of two fundamental theorems from the theory of con- 
tinuous functions are found in the next two theorems. 
Theorem 3. // fn{^) {n = 1, 2, . . .) are Ci(^(o)) while also 
lim„=oo/n (^) = / (^) uniformly for ^ in R, then f is Ci 
Theorem 4. // / (^) is Ci at all points of R, then \ f (^) | is finitely lim- 
ited and f attains its upper and lower bounds in R. 
As a foundation for much of the work in the sections on differential 
equations and imphcit functions, there is established for infinitely 
many variables an analogue of Taylor's Theorem with n terms and an 
