Suppose represents the set of all the functions from to ;that is, the set of all real-valued functions of a set of a real variable. Prove that multiplication is distributive over addition for .
What this means is:
Let , and are three functions.
We have to prove that(part 1) and (part 2)
Now from calculus we know that for two functions and :
Proof of part 1:
Proof of part 2:
So has multiplication distributive over addition.[Q.E.D]
The steps I've taken to prove this are they correct? Am I right to prove it like this? Can anyone kindly confirm?