Outline of a Theory of Functions of an Abstract Variable 63 
to a definite value of m, belong to a space E,n). It is not diffi- 
cult to show that for a,,) to have a norm, it is necessary and 
sufficient that @[njX%1-%2-°-* Xn, on Eq to E,o) be continuous in 
all of its arguments. One may then easily verify by reference 
to the postulates P.1 to P.14 inclusive that the following theo- 
rem is true. 
T.6. The space EF, is a complete linear vector space.* 
The extension of the operation of multiplication to the 
spaces E;,; necessitates a restriction, which is one, however, 
that will not limit the theory in its intended development. The 
Product @{mjb[nj, m Zn, is obviously to be defined as the aggre- 
gate of products dm)(x1, X2,° °°, Xm) *Deny(Vi, V2, °° * > Yn), With 
one restriction that the product is to be limited to an aggregate 
that depends upon (m+n—2) parameters at most. However, 
in order that the product satisfy the conditions (1) and (2) when 
these are extended for several variables, in addition to the con- 
dition of symmetry, it is also necessary to impose the restriction 
that multiplication is only to be allowed between elements @;,) 
and bin) when 
Gini (X1, Se somes ee Xm) *Btny(¥1; cepts (ie ae Vn) 
= O(mj(%1, * > * PES tetas te Bi oy Pio » Vn) 
Having now satisfied the conditions (1) and (2) extended in ad- 
dition to that of symmetry, it follows that there exists an ele- 
ment in Fimsn—2} which is equal to the product @{nj){nj. We 
must be careful to note that any two elements, one in Ejm),and 
one in E,,), m=n, are not multiplicable, so that whenever a 
Product appears in what follows, it’ will have been tacitly as- 
sumed that the product has a meaning, in other words, that the 
elements involved are multiplicable. 
Let us suppose that we have a function of the sort @[nj)X)(n} 
on En; to E(m+n—2}, With @m; and bn; both multiplicable ele- 
ments. For definiteness, let us take m= 3, = 2, say. Multi- 
plying on the right by yand then by z, we may consider the prod- 
uct, according to the associative law, as given by 31x { biayy Ie. 
Because of the symmetry, we have 
ats)x{biyy}s = ays {bia y} x2 
= { ajsybi2} yx 
= Ci3]+ 2, 
* It is apparent now that Eyn) is essentially a subspace of Ej,); i.e., the 
subspace of vectors all of whose components are equa 
