10 pitcher: interrelations of 



'$fls, the class W extended as to the class ®, is the class' of all func- 

 tions d of the form: 



e = J^fin (P;<t), 



n 



viz., of all limit functions of sequences {ju„} of the class Wl converging 

 on '^ uniformly as to some function (7 of 2. In symbols: 



n 



6. Closure properties. — In the theory of Moore certain closure 

 properties of classes of functions are fundamental. Of these closure 

 properties we will be principally concerned with 



A (absolute) ; L (linear) ; C (closed) . 



Classes possessing these properties are denoted by 



m""; »J^; m'', 

 respective!}'. 



(1) w^ -.^-.'laUfx .^. (A/x)^°^t 



viz., the class 50? is absolute in case 9}? is such that for every n (it is 

 true that) the function An belongs to 9)f. Thus the class of all con- 

 tinuous functions is absolute since the absolute of a continuous fuiiction 

 is a continuous function. 



(2) 50?^ : = : 50? 9 Ml • M2 . oi . Co . 3 . (ai^i + a.Ai.)^"'", 



viz., the class 50J is linear in case for any ^i and ;U2 of 9)? and any real 

 numbers ai and a2, the function Oiyui + 02^2 belongs to 9K. Thus the 

 class of all convergent series is linear. 



(3) w.^.m3Ti^, = m, 



viz., SO? is closed in case 90? is such that the extension of Tl as to itself 



* Here the form of is the same as in the definition of SKo-. In the former case, however, 

 the definition is relative to 33! and o- while here the definition is relative to 3}1 and S = [o-] 

 and accordingly the two forms have different significations. In the latter case the <r is a 

 part of the arbitrariness inherent in the notion /orm just as is the sequence .Vr.' in both 

 cases. 



t Relative to elements and classes in general an element i belonging to a class ^ is 

 denoted by the notations: i*; 1*0*, so that the notations p; p*; p^o* are interchangeable. 

 Similarly a class 3E of elements belonging to a class $ is denoted by: X*; X*"*. Belonging 

 to 5p is a relation between an element x or a class X of elements and the class 5P; but in 

 practice it is convenient to think of it as a property of the element x or class X of elements. 

 See footnote to § 4. 



