Odd and even functions- a direct sum? Question: 
Let V be the vector space of all functions $\Bbb R\to \Bbb R$.
Show that $V=U \oplus W$
for $U=${$f | f(x)=f(-x) \forall x$}$, $W={$f | f(x)=-f(-x) \forall x$}
What I did:
I did prove that $U \cap W$={$0$}. But proving that any function from R to R can be displayed as a sum of odds and evens wasn't a success. I tried saying that for $v \in V, w \in W: v=v-w+w$ and proving that $v-w \in U$ but that didn't work (That trick worked with some linear transformations we saw, but this isn't a linear transformation).

Hint: $$f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$$
