Every function is the sum of an even function and an odd function in a unique way

Question

It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However

How do you prove that this representation is unique?

Thanks for your help.

Answer 1

If $f = g_1 + g_2 = h_1+h_2$ where $g_1,h_1$ are even and $g_2,h_2$ are odd then $$ g_1 - h_1 = g_2 - h_2 \tag{1} $$ where the left-hand side of $(1)$ is even and the right-hand side is odd, hence both sides are just $0$. Indeed, it is easy to show just from the definitions that any function which is both even and odd must be a constant $0$.

Answer 2

An alternative way is via linear algebra. Let $V$ be the vector space of functions $]-a,a[\to\mathbb{R}$, and define the "reflection along the $y$-axis" operator $Y:V\to V, f\mapsto [x\mapsto f(-x)]$.

I'll leave it to you to show that $Y$ has precisely two eigenvalues, namely $1$ and $-1$. The eigenfunctions of the former are precisely the even functions, while the eigenfunctions of the latter are precisely the odd functions. Thus we have a splitting into eigenspaces $V= E\oplus O$.