398 REPORTS ON THE STATE OF SCIENCE. 



Pincherle has made considerable use of a very suggestive analogy in 

 which the functions of a field are compared with the points of a space of 

 an enumerally infinite number of dimensions and distributive operations 

 with linear transformations of this space into itself. The properties of 

 adjoint operations are then indicated by the correlative properties of 

 points and linear manifolds of order n — 1 in a space of n dimensions. 

 The use of a space of an enumerably infinite number of dimensions has 

 been further developed by Frechet ' and Schmidt. 2 



In some cases the conditions that two operations 



A(/), Ml>) 

 may bs adjoint can be expressed in a very simple form such as 



b ~o 



[A(f)f(x)dx=^A(f)f(x)dx 



a « 



or 



the latter form occurring in the theory of linear differential equations. 

 The other form shows that the operations 



6 



A(/) = Ux,t)f(t)dt, A(</>) = U{s)K(s^)ds 



are adjoint. 



An important property of adjoint operations is that if one operation 

 is composite, e.g., A = BC, the adjoint operation is also composite, and 

 consists of the adjoint factors in the reverse order, i.e., A = CB. 



In some cases an operation X can be inverted in a unique manner 

 by an inverse operation X"" 1 . The discovery of the operation inverse to 

 a given one depends on the solution of a functional equation: if the 

 operation involves a definite integral the equation will be an integral 

 equation. Since the operation inverse to a distributive operation is also 

 a distributive operation, it is convenient to have a general expression 

 for a distributive operation. 



This has been supplied by Hadamard in the case of the particular 

 class of distributive operations which are continuous ; these are called 

 linear operations. An operation is said to be continuous 3 in a field of 

 functions M if, when a(x), a;(x), a s (x) are a sequence of limited 

 functions belonging to M, and such that 



o n (x) -> a(x) as n -f co 



except, perhaps, for a set of values of x whose content, is zero, then 



A(-) =lim A((.„). 



n -» co 



' Rend. Palermo, 190G. a Mid., 1907. 



3 Bourlet, Annates dc I'Ecolc Nor in ale, 1903, scr. 3, t. 14. See also Hadamard t 

 paper. 



