I was doing some homework problems and was proving several facts about vector spaces (and was quite proud at how easily I was doing it) and suddenly noticed my proofs were circular. So I picked one to prove from scratch. I did it, but I want someone to verify that it is, in fact, a proof. Here it goes.
In the following, V is an Abelian group and F is a field. We have a defined function: defined as scalar multiplication. Under these assumptions, we may call V an F-vector space.
Proposition: Given and v any vector in V, show that .
Proof: I will assume that there exists an such that for all . Existence of such an f will be demonstrated.
Now, there exists an element 1 of F such that 1 is the multiplicative identity (F is a field). We have that 1f = f (for all f) for multiplication in F. Thus:
(The last step is a part of the definition of the scalar product.)
We defined above that . Inserting this in the first and last steps of the previous line we find:
By comparing this result to we see that the required f = 1. Therefore such an f exists and the theorem is proved.
(End of proof.)
It just seems too...convenient I guess...to assume an f, then basically cancel the darned thing out, then state what it is. I can't find an argument against the above being a valid proof, but I'm uncomfortable with it.
Any thoughts? Thanks!