Vector Space confusion For each of the following, I need to decide if it is a vector space over $\mathbb{R}$.  (You may assume that the set of all real valued functions on the interval $[-1, 1]$ is a vector space with the operations $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x).$
A=$\{f:[-1,1] \to \mathbb{R}:$ f is an even function $\}$ 
B= $\{f:[-1,1] \to \mathbb{R}:$ f is differentiable and $f(0)+f'(0)=1 $}
C=$\{f:[-1,1] \to \mathbb{R}: f(0) + \int_{-1}^1sin(x)f(x)dx=0$  $\}$ 
My question is (forgive my lack of experience): Do I really need to prove all axioms of a vector space hold? Or only show that closure properties hold? I'm clueless as to how to prove these are vector spaces. If anyone can help me with at least one of them, I can try the rest. Thank you.
 A: Linearity isn't sufficient in general to prove the sets are vector spaces, but it is necessary. Since you are given the parent set is a vector space, you simply have to check closure under addition, closure under scalar multiplication, and containment of the zero-vector. This is called the subspace test. I would try and find where things break and go from there.
For a vector space, we have closure over addition. So let's look at $B$. What happens if $f(0) = 1$ and $f^{\prime}(0) = 0$. Now take $g(0) = 0$ and $g^{\prime}(0) = 1$. Clearly, $f, g \in B$. So if $B$ is a vector space, then $f + g \in B$. However, $(f + g)(0) = 2$. So closure under addition fails.
For $A$, let's look at addition. An even function is a function such that $f(x) = f(-x)$. So if I take two even functions and add them, will the result be even? Are the functions commutative over addition? What about associative? You may find properties of addition over $\mathbb{R}$ to be helpful here. By this, I mean that since the functions map to values in $\mathbb{R}$, it suffices to consider commutativity, associativity, and distributivity over $\mathbb{R}$. 
Now is there an identity in $A$? Is there a function such that for all $f \in A$, there exists a $g \in A$ such that $f + g = f$? Again, think about the additive identity on $\mathbb{R}$. 
Hopefully this will help you get started. I've included more steps than necessary, but I did so in the hopes of clarifying general vector space proofs as well, rather than restricting to the case of subspaces (which you should do here). Please let me know if I can clarify some.
