PROPERTIES OF CLASSES OF FUNCTIONS 11 



is the same as 93?, i. e., 9JJ contains the limit functions of all sequences 

 inn] of 9)J which converge uniformly as to some function /x of Wl. 

 We note two other closure properties and a proposition. 



(4) gj^muitipiica.iv. a' '0 " : ^ : grij , ^ . a . 3 . (a^)^"'-". 



(5) sK'"i'iitive_ : gr)j , ^, . ^. . ^ . (^1 + ^^y^^, 



/(W ffl?-^ r\i Orpailiiitive . multiplicative as to ?i 



viz., that the class W is linear is equivalent to (implies and is implied 

 bjO the class 9}J is additive and multiplicative as to constants. 



7. Dominance properties * — The dominance properties of classes of 

 functions with which we shall be concerned are D, Di. 



A class 9}f of functions has the dominance property D in case for 

 every sequence jm„| of functions of 9JJ there exists a function fi of 9}? 

 and a sequence {a„} of real numbers such that for every n the function 

 yun is dominated by the function n„/i. In symbols: 



93r . =. 9?? 3 { \^ln] .3.3 (o» . ju) s^M.p ^ ^a./^P (np)). 



A class 9}? of functions is said to have the dominance property Di 

 in case for every two functions fii and 1x2 of 9Jf there exists a number a 

 and a function /x of 9JJ such that the function an dominates /xi and 1x2- 

 In symbols: 



9[»r' . = . 9JJ 9 (mi . M2 . 3 . 3 (a . m) 3 ^4^1 ^ Aafi . Am ^ Aa^). 



The following propositions are holding: 



(1) 9[)r.D. 9JJ^'; (2) 9JJ-" . 3 . 9)^'. 



The dominance properties are fundamental and especial importance 

 attaches to the dominance property D. This dominance property 

 enters the theory chiefly as a property of classes functioning as scales 

 of uniformity. The reader is referred to the Introduction to General 

 Analysis, §§21-26, for discussion of the various dominance properties. 

 We state Theorem I of §25 of the above-mentioned work. 



Theorem I. If a class <£■" is the scale class of each of a sequence of 

 instances of relatively uniform convergence, then the corresponding 

 sequence {(r„ } of scale functions may be replaced by a single scale function 

 a effective for each of the instances. 



A similar theorem may be stated for Di. 



* I. G. A., §§ 19; 22. 



