Outline of a Theory of Functions of an Abstract Variable 65 
ll ocny(a>|| Spa 
issatisfied. Evidently f(x) is continuous on S;, because Any (X) +x 
is, and the sequence {S:} has for its outer limit the space E,). 
Hence, from the definition, f(x) is an M-function. 
By an easy generalization of T.7. it is possible to deal with 
the case of several variables. Thus, we have 
T.9. Tf f(x, x2, +++, %,), a function on Ba” to Bn, 
linear continuous homogeneous in each variable, and uniformly 
so for all values of the remaining ones, and if, moreover, it is 
Symmetric in every pair of elements, then there exists an ele- 
ment @(n) of E,,) such that 
T=, ee %,) = A[njX1%2- + * Np. 
Proof. If we consider f(x, x2,-- +, x,) as a function of x, 
alone from the previous theorem we have 
fi 41, Ya, ++, Xr) = A[n—r41] tr 
for all values of x,, where @,,_;+1) is a function of the remaining 
variables, which has the same properties as the given function. 
Considering any particular value of x, and @,,-,+1) as a function 
of x,; alone, we again reproduce the conditions of T.7, so that 
we may write 
Gn—rp1](H1, Hay ++ + Mr) = O[n—rp2]%r—1- 
This holds for the particularly chosen value of x,, and on consid- 
ering the aggregate of all values of x,, we obtain an aggregate of 
vectors containing @1,_;42;.. This aggregate, however, defines a 
single vector in E;,_,42; with one more variable component 
than @n_,,9)}. Then, on carrying out the procedure of complete 
induction, we have finally 
ff .%1, Xa, - + * 5 My) = Bn tite* «Are 
From the continuity of the function, it may be seen that the 
horms of the coefficients successively determined are well-de- 
fined, and consequently that 
S {lanl l]al] «- - [loa 
I[far, 2, +++, 2) 
Polynomials.* A function on Eq) to E,,-rj, of the form 
O[n\ x" + One"? + sae + @(n—rt 1} ¥ 4 Qtn—r] 
is called a polynomial of degree 7, provided, of course, that @[n) 
* For other definitions of abstract polynomials see the note at the end 
of this paper. 
