64 Transactions of the Academy of Science of St. Louis 
where ¢€(3}= 4 3}b~). Accordingly, it follows that 
G31 Xb 2) = C13)%, 
and generally, that there exists a sort of commutativity among 
the factors of a product which are multiplicable among them- 
selves. 
It is also easily seen that if two vectors are multiplicable, 
and the second is multiplicable with a third, then the product of 
the first two is multiplicable with the third. Obviously, multi- 
plicability is immediately extensible to the sum and scalar prod- 
uct of a set of multiplicable vectors. 
For completeness EF 19; is defined to be identical with Eq). 
It is a simple matter now to verify the fact that the spaces 
En) satisfy, with the exception of P.20, all the postulates for a 
generalized vector space 
General multilinear forms. We have the following theorems. 
T.7. The most general linear continuous homogeneous func- 
tion on Eq) to Eqs is a function of the form @,,)x, where @{n1 
is an element of the space E4y). 
Proof. Because of P.20, for a fixed value of x, say Xo, we 
may solve the equation 
U(nyXo = f (0) 
for un), where f(x@)) is the given function which is linear con- 
tinuous homogeneous. For all choices of x we obtain a set of 
elements 1 ,)(x) that have the properties (1) and (2). Hence 
the aggregate 1,,)(x) defines an element ;,}, which, since f(x) is 
continuous, belongs to the space E,,;, being a vector, all of 
whose components but one are equal. 
T.8. An arbitrary linear homogeneous function on Eq) to 
E(n—1) which may or may not be continuous, is an M-function. 
Proof. As in the previous theorem, if f(x) represents the 
given function we can find an aggregate of elements in Ey), 
namely d,,)(x), which satisfies (1) and (2), so that 
@(n)(%) eS f(x) 
for all x. Obviously the norm of a,,)(x) is a function of x, which, 
because f(x) is not assumed continuous may not be pounded: 
et {ui} be a sequence of positive numbers tending mono- 
tonically toward infinity. Let S; be the set of points in Eq 
for which the inequality 
