MATHEMATICS: G. D. BIRKHOFF 
657 
this event, if we multiply through the equation of definition for E by 
f{x) and integrate as we may by hypothesis, we find 
J^V W H y) dx=^ Jo'/ ^« ^^'"^ 
«= 1 
where the series on the right-hand side will converge uniformly by 
hypothesis and so represents a continuous function. 
This series on the right has the value f{y) . In fact the difference 
n= 1 
is a continuous function (^^(y) such thsit I <p (y)u„(y) dy vanishes for all 
precisely because the set Wi(ic), W2(^), • . . is normalized orthogonal. 
But the set Ui(x), u^Xx), ... is closed by hypothesis; hence we infer that <^ 
vanishes identically. Hence the right-hand member of the preceding 
equation has the value /(y). That equation may now be written 
^lf{x)H{x,y)dx^f{y). 
If the maximum numerical value of / (x) for 0 ^ x ^ 1 is F 4= 0, and if 
this value is taken on for y = yo, we get 
\^'j{x)H{x, y,ydx\=F. 
But this is impossible since by hypothesis | // | < 1 and \ f \ -^F. 
The set Ui{x), u^ix), ... is therefore closed also. 
2. An Application. The Sturm-Liouville series, in a specialized but 
typical form, arises from the set of functions Ui{x), u^ix), . . . which 
are the solutions of a linear differential equation of the second order 
+ (X - g{oc) ) w = 0 
satisfying boundary conditions u{0) = u{l) = 0, for the ordered set of 
parameter values Xi, X2, . . . of X respectively. We will assume that 
g{x) and dg{x)/dx are real and continuous. 
By means of the methods of Sturm it is proved thatwi(x),W2(x), . . . 
may be taken to form a normalized orthogonal set in which w„ {x) will 
vanish precisely n times within the interval 0 ^ x ^ 1. And the meth- 
ods of Liouville give the asymptotic form of u„ (x) ; it will be convenient 
for us to use the formula 
/ N /-f • I <^Wcos«7rx , M„{x, g{x))'\ 
u„{x) = -y/ 2\ sm nirx + — ~ + — ^ 
L n A 
