Why can any wave function be written as the sum of parity eigenstates?

Question

I read in Zettili's Book of Quantum Mechanics that any wave function $\psi(\vec{r})$ can be written as a sum of the parity operator's eigenfunctions, since its eingenstates $ |\psi_+\rangle$ and $ |\psi_-\rangle$ form a complete set, then $\psi(\vec{r}) = \psi_+ (\vec{r}) + \psi_- (\vec{r}) . $ But I still can't see why. Could you help me, please?

Answer 1

Any function $f(x)$ can be written as the sum of an even function $f_+(x)$ and an odd function $f_-(x)$, $$f_+(x) = \frac{f(x) + f(-x)}{2}, \quad f_-(x) = \frac{f(x) - f(-x)}{2}$$ because $$f_+(x) + f_-(x) = f(x).$$ This is just the generalization of that to three dimensions.