MATHEMATICS: C. A. FISCHER 
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and since from definition 
the right member of the last equation cannot be greater than M. That is 
Vzu (5, /) ^ M. (5) 
It can be proved by the method Riesz uses in showing that a linear 
functional is defined uniquely for the limit of a sequence such as (4), 
that the function u (s, t) just defined is the limit of u(s-{-e, t) when e = 0. 
Thus u (s, t) is regular. 
It will now be proved that 
U{f,g) = ^^I{s)g(.l)d,u{s,t), (6) 
where / {s) and g (t) are arbitrary continuous functions. The function 
/ ' (s) will be defined by the equation 
/' (s) = ^/(s'd (/ is, sl+ ,) -lis, S-) ), 
i 
and (Q in the analogous way. Then the equation 
u (/; = 2 / ^''^ s (9 « n (7) 
will be satisfied. It follows from definition that f'(s) = /(^J) in the 
region Si< s ^ s\+i, and similarly for g^ and g. Since U is linear in 
each argument and / and g are continuous, U (//, g') approaches U(f, g) 
when the length of the greatest of the intervals approaches zero, and 
the right member of equation (7) approaches the Stieltjes integral in 
equation (6). Inequalities (2) and (5) imply that V2U is equal to M, 
This completes the proof that when U (/, g) is bilinear there is a unique 
function u(s,t) which is regular and satisfies equation (6), and the 
variation of u is the least upper bound of | U(f, g) | /mfmg. 
If this property is assumed for functionals linear in each oin — 1 argu- . 
ments, the proof just outlined can be modified to make it prove that 
functionals linear in n arguments have the same property. Thus the 
theorem holds for /^-linear functionals. 
This can be used to extend a theorem of Frechet's^ about functionals 
of the second order to those of the nth order. A functional of the nth 
order is defined as one that is continuous and satisfies the equation 
