[Math] Proof of subspaces of odd and even functions

Question

1. $F^+(\mathbb{R})$, the set of even functions in $F(\mathbb{R}, \mathbb{R})=\{ f: \mathbb{R} \to \mathbb{R} \}$ and $F^−(\mathbb{R})$,
   the set of odd functions in $F(\mathbb{R}, \mathbb{R})$ are both subspaces of $F(\mathbb{R}, \mathbb{R})$.

2. $F(\mathbb{R}, \mathbb{R}) = F^+(\mathbb{R}) \oplus F^−(\mathbb{R})$.

How to prove the statements above only using the definition of subspaces?

Answer 1

1. Write down the definition of subspace and check that $F^+(\mathbb{R}),F^-(\mathbb{R})$ satisfy the properties. This should be trivial.

2. Suppose $f: \mathbb{R} \to \mathbb{R}$, then $$f(x) = \left( \frac{f(x)+f(-x)}{2} \right) + \left( \frac{f(x)-f(-x)}{2} \right)$$ Can you show that if $f = h^+ + h^-$ is another expansion of $f$ that $h^+(x) = \left( \frac{f(x)+f(-x)}{2} \right)$ and $h^-(x) = \left( \frac{f(x)-f(-x)}{2} \right)$? Hint: consider differences.