642 
MATHEMATICS: C. A. FISCHER 
approaches zero. The details of the proof of this will not be given 
here. 
If two regular functions u {s, t) and u' (s, t) satisfy the equation 
Jj,/ W g (0 d,u (s, t) = j^f (s) g (t) E^u' is, t) (3) 
identically in / and g, they will be proved to be identical. Since the 
similar theorem has been proved for single Stieltjes integrals, if u and 
ti' are unequal at a point {s' , t') there must be a continuous function 
^g{t) such that V (g; s') 4= "v^g; s'), where is the functional analogous to 
v(g;s). Then since v and are regular in s there must be a continuous 
function / (s) for which the two members of equation (3) are unequal. 
This proves that if a functional U (/, g) satisfies the equation (1) where 
u (s, t) is regular and independent of / and g, the function u {s, t) must be 
unique. 
As Riesz has proved,^ the field of functions for which a linear func- 
tional is defined can be extended to any function which is the limit of 
a sequence of continuous functions which satisfy the inequalities 
/l^/,&/3& . . . , (4) 
and any linear combination of such functions. Thus if U (/, g) is bi- 
linear, one or both of the functions / and g may be discontinuous if it 
is the limit of such a sequence. A function / {s, s') will be defined by 
the equations 
a) = 0, 
}{s,s') = \, (a ^s';s'>a), 
f{s,s') = 0, {s'<s^a'), 
and a function g (t, t') by the analogous equations. If / is considered 
a function of it is approached by a sequence such as (4), as Riesz 
has shown,2 and g (t, t') has the same property. The function u (s, t) 
will then be defined by the equation 
u{s\t')= U{f(s,s'),g{t,n)- 
This function vanishes for =a, or t' = h' , by definition. Since U is 
bilinear the expression U {f, g) /mjmg is bounded^ and its least upper 
bound will be called M, a constant independent of / and g. The varia- 
tion of u is defined as the upper bound of the expression 
2J e^e^ A,, u {s\ = U (^e,{f (s, sl+ ,) -f {s, si) ^ ej (g (/, /J+O - g (/, //) )), 
