How to verify that $\mathbb R^\mathbb R$ is a direct sum of the Ue and Uo

Question

The $U_e$ and $U_o$ denote the set of all real-valued even/odd function on $\mathbb R$ respectively.

Which bits are you struggling with? Can you show that $\mathbb{R}^\mathbb{R}$ is isomorphic to the functions from $\mathbb{R}$ to $\mathbb{R}$? — Alexander Cutbill, May 04 '15 at 08:01

score 2 · Accepted Answer · answered May 04 '15 at 08:04

Have you done the following exercise in matrices? Every matrix is the sum of a symmetric and an anti-symmetric (i.e. $A^T=-A$) matrix.

Define $f_e(x) = \frac12\big(f(x)+f(-x)\big)$ and $f_o(x) = \frac12\big(f(x)-f(-x)\big)$. Now $f=f_e+f_o$. Leave it to you to check the sum is direct, that is uniqueness of this expression.

How to verify that $\mathbb R^\mathbb R$ is a direct sum of the Ue and Uo

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