Colloquium publications.

FUNCTIONALS AND THEIR APPLICATIONS.

109

variable p in A, to which the functions r and ~ of equation (G)
belong; 1 is the class of functions of t on which the operation J
may be performed; and finally, 9 denotes the class of functions
to which the kernel function K(st) of (G) must belong.
The significance of the equations
(12) 9 - 9 *2
and
(13) 9 = (9)if)*
may now be made clear. The symbol T)2 denotes the class of
product functions Il(p)g2(p) of a single variable, of which the
factors are functions of 9. On the other hand, the symbol
(TO9i ) denotes the class of product functions l1(Pl)/J2(P2) of two
variables, of which the factors are functions of V); in this latter
case, pi and p2 are independent variables, each one ranging
over $.
A symbol ~9* denotes the class of functions obtained by
adding to P), first the functions
f = ailp + 22 + * * * + a.nA,
the functions /i being arbitrary functions of 9), and the coefficients ai being arbitrary numbers from 21 (i. e., arbitrary real
or complex numbers as the case may be), thus forming the class
designated as T9L, and second, the functions
(14) /Up = lim fig,
i=-oo
where {,i} is a sequence of functions p which converges uniformly to gp, over 13, relatively to a scale function which must be
taken from V9/L. It can easily be shown that if 9) possesses the
property (D), given below, then 9NJL and T*, possess the same
property, and uniform convergence relative to the class V) is the
same as that relative to V)1L or 9)*. Moreover (9,*)* = 9)*.
Hence the significance of (12) is that W9 is the totality of functions p(t) such that

(12')

v(t) = lim vi(t), (V; V),
i=oo