Mathematics. — “A general definition of uniform convergence with 
application to the commutativity of limits”. By Prof. Frep. Scnun. 
(Communicated by Prof. CARDINAAL). 
(Communicated in the meeting of May 3, 1919). 
1. Let us assume two aggregates V and W with coverings of 
the positive numbers by the parts of V resp. W, as described in my 
paper “A general definition of limit’, p. 46. By these coverings to 
a positive number J a part V‚ of V and a part W; of W is made 
to correspond in such a way, that Ve and Wes contain al least one 
element, Vs being a part of Vs and Wea part of We if I <4. 
Let VW be the productaggregate, whose elements are formed by 
combining an element of V with an element of W into a pair 
(disregarding sequence). The covering by the parts of VW is taken 
such, that the aggregate Vs Ws corresponds to J. 
If we replace the coverings belonging to V and W by equivalent 
ones (c.f. n°. 3 of my previous paper) the covering belonging to 
VW is likewise replaced by one equivalent to it. 
2. We assume VW covered with (real or complex) numbers, so 
that every element of VW is made to correspond to a number. 
From this arises an aggregate K of numbers, in which the same 
number may occur several times. We shall indicate by G an aggre- 
gate of elements of K, corresponding to a same element of V, com- 
bined with all elements of W. There likewise arises an aggregate 
H by considering a constant element of W. 
The covering by the parts of VW can be transferred to the 
ageregate K. Likewise the covering by the parts of V or W is to 
be transferred to every aggregate H resp. G. 
3. Suppose that every aggregate G (as regards the covering by 
the parts of W) possesses a limit Lg; then corresponding to every 
element of V there exists anumber Lg. We say, that the aggregates G 
converge UNIFORMLY to their limits Lg, if corresponding to every 
positive number « there exists such a positive number d, that each 
element ZE of the part Gs, of G satisfies the inequality |A—Lg| <e, 
whichever element of V (hence whichever aggregate G) we may 
choose ; it being required, that d is independent of the chosen element of V. 
