FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 119 



converges absolutely and uniformly, and has for its sura function K x (s, t), the solving 

 function of K (s, t). 



Let /(*) be a function which has a Lebesgue integral in (a,b). Since (1) is 

 uniformly convergent in the square a S * ^ b, a : t ^ b, it is clear that the function 



r = 1 



<V""~ *. 



satisfies the requirements of the theorem of I., 3, for these values of s and t. We 

 deduce that 



and that the series on the right converges uniformly for values of s in (a, 6). 



It is easy to show directly that the series on the right of (2) is absolutely 

 convergent. The result, however, follows at once from the fact that we have 

 throughout left the order of the terms of the series arbitrary, which would otherwise 

 have been impossible, by KIEMANN'S theorem on derangement. 



2. We may here digress to prove a slight extension of the Hilbert-Schmidt 

 expansion theorem,* applicable when K (s, t) has the properties mentioned above. 

 Writing X = in (2) we have 



f K (8,t)f(t)dt= 2 *(*) f 



J a = 1 A, .' a 



Now, the function K (s, t) / (s)f(t) has a Lebesgue integral in the square a f 6, 

 n^t^b; further, the repeated integrals 



f ' fr (s) ds f K (s, f(t) dt \"f(t) dt\" K (s, t) ^ (*) ds . 



Jo Jo Jo 



have a meaning. It follows from a known theorem t that the latter are equal ; and 

 hence that, as 



f j/ () V. (t) dt = -\" i/. (0 /() *,' where <j (*) = ( K (s, t) f(t) dt. 



fm A J 



Supplying in (3) we see that 



* HILBERT, 'Gott. Nachr.,' 1904, pp. 73-75. SCHMIDT, ' Math. Ann.,' vol. 63, pp. 451-2. 

 t HOBSON, 'The Theory of Functions of a Real Variable ' (Cambridge, 1907), p. 582. 



