PROPERTIES OF CLASSES OF FUNCTIONS 31 



viz., if 9}r has the properties K[^, K[,^ then W is such that for any 

 m"'''"' it is so that (WW)^ has the property K'M'm")- 



(4) ^rK,'.Kw. _ _^ _ gjj, ^ (gjj"i^« . 3 . (gofj'3)j")/'"™'<™">). 



(5) »?'''■'* . 3 . 93r 9 (502"^''° . 3 . (9}?'9}r')/='™''""' : ~ : SQ?"""'"). 



(6) m''"''- . 3 . 9jr » (2)?"^'''' . 3 . (gjrgjr')/''"''"""' : ~ : aR'""*^'*). 



(7) gjj,o.AVAV. _ ^ _ ^,A-„'._ 



(8) gp,j/^^='^V. _ ^ ^ gr,j,ic..'._ 



(1, 2, 3, 4) follow from direct consideration of the definitions of the 

 properties K[^, K'.,^, ivjo^ and the classes of functions involved in these 

 definitions. (5) and (6) follow from (1, 3) and (2, 4) respectively. 

 (7) follows from (2) and proposition (6) of § 29. (8) follows from 

 (1) and proposition (9) of § 29. We state (5, 6) and (7, 8) respec- 

 tively as the following theorems : 



Theorem I. // a class Tl' of functions on ^' has the property ki', 

 relative to a development A', then the statement that (SDiJ'SD?")* (S)f" an 

 arbitrary class of functions on '^" possessing the properties LCD) has 

 the property xi'si'S'sl") is equivalent to the statement that W has the property 



Theorem II. If a class 9)? of functions on ^ has the property °', 

 and the properties 5,', 5,* relative to a development A, then the class 9}J has 

 the property i^i2*. 



31. It is not difficult to see that whatever the class 'i)? and its de- 

 velopment A there exists a developmental system X> = ((S"*')) such 

 that conditions la and I'b of § 17 are holding. Furthermore, it is 

 clear that the class !D is bounded by the class of constants. These 

 considerations give rise to the following propositions relative to a 

 class "^ and a development A: 



(1) 9« 9 [all m''''"T"" • 3 . 9[)P, 



viz., if 9}J is such that all functions /x, which are dominated by the class 

 of constants on ^, belong to 9}J then ')})} has the property A. 

 The following propositions then follow in order: 



(2) 9K 95? = [all ix""'"] = [all m^'="] • 3 • [all ;u*'='<*f "'« . m\ 

 (3 9)?^-^-. 3 . [all ;i^.''(*)]^»™- . m^\ 



