The number of spaces may of course be arbitrary. If, in addition, we agree to identify vector spaces whose bases have the same cardinality, the operation of direct sum will become commutative as well. Elsewhere we looked into the direct sum of Boolean Algebras. If Y is a set of numbers, the function is called numeric . Numeric functions can be added.
X R be two real functions.
The value of the sum is the sum of values. Note that, in general, for an arbitrary function f , its value at one point does not depend in any way of its values at other points. The sum of functions is also defined pointwise . Therefore, whatever is true for individual points also holds for the whole function. Associativity is shown in a similar manner.