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?

  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.


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