26 
MATHEMATICS: C. A. FISCHER 
Proc. N. a. 
From (12) wc have 
and (12) becomes 
Rpr = \ gprR- (13) 
The original Einstein equations (1914) for space free from matter are 
those for which <^ = 0 in (12); in 1917 {Sitz. Pr. Ak. Wiss., Feb. 8), those 
for which (p = const.; and in 1919 (Ibid., Apr. 10), the general case (13) 
of a homogeneous space from the above point of view. 
From (8), (9) and (12) it follows that for any three mutually orthogonal, 
directions in any 3-space, which is homogeneous, 
ri2 + ri3 = r2i + f23 = rsi + ^32 = — (p, 
whence 
ri2 = ri3 = f23 = — ^• 
Thus the Riemann curvature at each point is the same for all directions^ 
and by the theorem of Schur (Math. Ann., 27, 563) is constant. Conse- 
quently the first type of Einstein space is a generalization of euclidean. 
3-space, and his other two spaces of 3-space of constant curvature. 
NOTE ON THE DEFINITION OF A LINEAL FUNCTIONAL 
By Charlks Albert Fischer 
Department oi? Mathematics, Trinity ColIvEge, Hartford, Conn. 
Communicated by E. H. Moore, January 5, 1922 
A linear functional is usually defined as one that is distributive and 
continuous, but the term continuous functional has been used in at least 
two ways which are not equivalent. F. Riesz, Frechet, G. C. Evans and 
others have defined a continuous functional as one which satisfies the 
equation 
limit L{u,,{x))=L{u{x)), (1) 
n — 00 
when the sequence Ui{x), u^ix), approaches u{x) uniformly, while 
Levy^ and W. L. Hart^ have simply assumed that the sequence of 
converge in the mean. In Levy's paper the functional also depends on 
a parameter which has the same range as x, and L{Un{x),y) is only required 
to converge to L{u{x),y) in the mean, but such a parameter will not be 
introduced here. 
In what follows a distributive functional will be called linear when 
equation (1) is satisfied for uniformly convergent sequences of w's, and 
linear-m when they are only required to converge in the mean. If then, 
L is to be linear it is necessary and sufficient that the equation 
L{aui + hu2) =aL{u\) + hL{u2) (2) 
