Proving a Vector Space I was not really sure where to start with this question so if someone would please just point me in the right direction that would be great:
Let $N$ and $L$ be vector spaces over the same field $F$. Show that $FCN(N,L)$ is also a vector space over the same field. 
FCN(N,L) is the space of functions form N to L, but we don't know that the function is a linear function.
 A: Defining, when $f$ and $g$ have the same domain (N in this case) and codomain (L in this case):
$(f+g)(x):=f(x)+g(x) ~~ \forall x \in N$
Note that the fact that L is a vector space is important, otherwise the right side of the equation could not be in L. Note also that the "sum" in the left side is the sum we are defining, and the "sum" on the right side is the sum of the vector space L.
Now, this makes $( FCN(N,L), +)$ an abelian group. 
Defining:
$(c.f)(x):=c.f(x) ~~ \forall x \in N$
With the proper observations, analogous to the first case, we have that $FCN(N,L)$ is now a vector space over $F$.
A: Then it's best to state the axioms of a vector space. The book or notes you are using should contain them and if not you will find them in the Wikipedia entry on vector space. 
For example one of the axiom states that there is a binary operation $+$ defined on $V$ such that $v_1 + v_2 = v_2 + v_1$ and such that there is a neutral element $e$ and such that for every $v$ there is an inverse element. To verify that this axiom is satisfied observe that if $f,g$ are functions $N \to L$ then their pointwise sum $(f+g)(n) = f(n) + g(n)$ is again a function from $N$ to $L$. 
You will also need to check the other axioms, such as scalar multiplication. I am somewhat reluctant to post a complete solution mainly because I think it's a good exercise for you to go through the axioms.  
