PROPERTIES OF CLASSES OF FUNCTIONS < 



S[R"" is the class of all functions n on ^]?"" to 21, viz., of all 

 n-partite real numbers : 



(III) 'iP'" is the class of a denumerable infinitude of elements: e. g., 



p = 1, 2, 3, •• ■, n, • • •. 



g[)J'" is the class of all functions ju on ^™ to 21, viz., of all in- 

 finite sequences of real numbers: 



M = (MpIp) = (mi, M2, • • •, Mn, • • ■)• 



(IIIo) 9)1"'° is the subclass consisting of all functions n such that 



Lmp = 0. 



P=CO 



(Ille) SOJ'"" is the subclass of all functions n such that 



converges.* 

 (IV) ^13''' is the finite interval! (0 1) 



^ p ^ 1 



of the real system number 21. 

 SfJJ'^ is the class of all continuous functions of p on "^^^ to 2(. 

 3. Relatively uniform convergence. — Moore introduces the notion 

 of relatively uniform convergence. % In order to introduce the notations 

 and clarify the idea we indicate here the various kinds of convergence 

 reserving until last the relatively uniform convergence.! 



The sequence {^r, I of functions tx„ on ^ to 2t converges to the 

 function B as limit : 



(1) Lj". = e, 



n 



in case 



(2) p . e: 3 : 3 ?ip, , > Upe . 3 . ^(m,.p — ^p) ^ e, 



viz., for every p and e|| (it is true that) there exists a positive integer 



* A denotes absolute value of. 



tThe theory relates equally to case IV for any interval (aocii)- 



X I. G. A., § 7. This is an instance of relative uniformity in general. See I. G. A., 

 §6. 



§1. G.A., §7. 



II e denotes a positive number greater than zero. The class of all such is denoted by [e\. 



