PROPERTIES OF CLASSES OF FUNCTIONS 29 



Consider classes: 



5i = [all ^"vi.'c™").«oi'!"(P')]. 



5 12 = [all ^^-'»''('^'") . ^«'""(J'')], 



which occur in the definitions of K[^; K[.,^ respectively. By (6) 



.-. gi = 5i2 and a)r*'''' . - . m'"''-'- 



28. A relation between the 'properties iv,^, ii,^, K^.,^. — If a class 

 M' of a system (21; %'; A'; W) has the properties K[j^ K'^^ then the 

 classes* gi and 52 defined relative to an arbitrary class 'JSl" , oc- 

 curring in the definitionsf of K\^ K'.,^, are each the same as the class 

 m''M")^. Since 



5i2 =n((Vig2), 

 we have 



5i2 = 5i = g2 = m'm")^. 



Theorem. — // a class Tl of functions of a system (31; '^; A; SOt) has 

 the properties K^^^, K^^ then it has the property Kij^.. In symbols: 



29. Propositions. — The following propositions J are relative to a 

 class ^, a development A of 'i)3 and classes Tl = [fi], dl = [p], 'Si = [a] 

 of functions on ^ to 21. 



(1) 9}?^'*-'' . ^Tf^'™ . (g^i'™ . 3 . 3J<,-'^''"'. 



(3) a«^'^' . 3 . 2«i^''^'=" . m.^''^"''" . TO*^'^'"'. 



The proof of (1) is straightforward. (2) and (3) then follow in 

 order by the use of (1) and (2) respectively, and by proposition (5) 

 of § 9a. We restate (3) as 



Theorem I. If a class 3)1 of functions of a system (31; ^^; A; W) 

 has the dominance property Di and the property Ki then the linear ex- 

 tension of W, the extension of 9)J as to itself and the ^-extension of 3)1 each 

 have the dominance property Di and the property \ Ki. 



* 5, = [all ^AV«!'(M"). J5o«"(>')] (i = 1, 2, 12). 



t§23. 



t Propositions 1, 2, 3, are to be compared with propositions 5, 6, 7, of § 23o. 



§ Cf. § 23a2i. 



