8 pitcher: interrelations of 



ripe (dependent on p and e) such that for every n exceeding Upe (it is 

 true that) the absolute value of iiinj, — 6p) is at most e. 

 The convergence is said to be uniform over the range "i}? : 



(3) LMn = e m, 



n 



in case 



(4) e : =) : 3 n^ 9 n > «„ ■ => . A(m„p — 0p) ^ e (p),* 



viz., for every e there exists a positive integer Uc such that for every 

 n exceeding iie the absolute of the function (fXn — 6) is for every p at 

 most e. 



For a function a, the absolute of a-, in notation : Aa/is the function on 

 ^ to 91 for which {A(t)p = A(a-p) for every p. 



The sequence {;u„} is said to converge to 6 uniform!}^ on the range 

 ^ relative to or as to the scale a: 



(5) L/xn -e {%v,a), 



n 



in case 



(6) e : 3 : 3 He i n > rie .z^ . A ()u„ — d) ^ eAa, 

 that is, 



(6') e :z) -.3)1^3 n > Ue .z). A(ijl„p — dp) ^ eAcTp (p), 



viz., for every e there exists a positive integer n^ such that for every n 

 exceeding n^ it is true that for every p the absolute value of (finp — dp) 

 is at most e multiplied by the absolute value of o-p. 



Uniform convergence is an instance of relatively uniform con- 

 vergence the scale function being the constant function 1. 



We speak of the convergence of the sequence j^„j on ^; uniformly 

 on "ip ; uniforinly on ^ as to a, in case there exists a function 6 effective 

 as the limit of the sequence : 



Lm« = e (p); Lm„ = e {%^); Lm« = e {%^; a), 



n n n 



for the respective modes of convergence, f 



3a. Relative uniformity as to a class of functions. — A sequence 

 {Hn] of functions is said to converge to a function 9 uniformly as to a 

 scale class © = [a], in notation: 



LMn = e (^;©), 



n 



* The significance of the (p) at the end of a statement is that the previous relations 

 hold for every p of the range ^ 

 fl. G. A., §8. 



