function spaces in linear algebra I am in upper division linear algebra, and i need help in proving a function space as a vector space. I just need help proving 2 particular vector space axioms.
Axiom 1: For each pair of elements a,b in F(field), and each element x in V (Vector space), ab(x)= a(bx).
Axiom 2:For each element a in F and each pair of elements x,y in V a(x+y)=ax+ay
To prove axiom 1, you would need 2 scalars, so can I just say, let a and b be a member of F... So wouldn't the proof be exactly the same as the axiom? 
ab(s)= a(b(s))?
To prove axiom 2: I really don't know where to start....
What are your tips and tricks to prove these axioms when you have function spaces? I am having a lot of trouble proving how the axioms work. Any help would be greatly appreciated...

It's generally a bad idea to try and work with a vector space when you haven't got a good idea of how it's defined. What do we mean by scalar multiplication? Addition of two vectors? The way it's usually set up is as follows:
If $V$ is a set of functions from $X$ to $Y$, where $Y$ is a vector space over a field $F$, then we define the addition of two functions:
$(f+g)(x) = f(x) + g(x)$ for all $f, g \in V$ and all $x \in X$.
and multiplication by scalars:
$(\alpha f )(x) = \alpha(f(x))$ for all $f \in V, \alpha \in F$ and $x \in X$.
Once you have understood the setup, it's pretty easy to show the axioms are fulfilled.
