Let $E,O \subset F(R,R)$ denote the sets of even and odd functions, respectively. Prove that $F(R,R)$ is a direct sum of E and O.

Question

Let $E,O \subset F(R,R)$ denote the sets of even and odd functions, respectively. Prove that $F(R,R)$ is a direct sum of E and O.

I proved that E,O are subspaces, and I can prove $E \cap$O={0}. How to prove that $U+O= F(R,R)$?

Answer 1

YOu simply need to show that every real-valued function is sum of even and odd functions.

Consider $h(x)=\dfrac{f(x)+f(-x)}{2}, g(x)=f(x)-h(x)$