I read awhile back that the set of continuous real valued functions from $\mathbb{R} \to \mathbb{R} $ has a direct sum decomposition into subspaces of strictly even and odd functions. Any such function $f$ could then be uniquely written in terms of even and odd parts by the formulas $\text{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $\text{odd}(x)=\dfrac{f(x)-f(-x)}{2}$ where $\text{even}(x)+\text{odd}(x)=f(x)$.
Recently I stumbled into a similar claim for $n\times n$ matrices, but instead of even and odd, the set of square matrices has a direct sum representation in terms of symmetric and skew symmetric parts. So given any square matrix $A$ we can write $B=\dfrac{A+A^T}{2}$ and $C=\dfrac{A-A^T}{2}$ as the symmetric and skew symmetric representations where $A=B+C$.
Does this mean that the notion of symmetric and skew symmetric have a connection to the notion of even and odd? Do they represent a generalization of even/odd, or is there another idea that generalizes these two ideas?
Let $V$ be any real vector space, and let $T$ be any operator on $V$ with $T$ not the identity but $T^2$ the identity. Given any $v$ in $V$, we have $v=u+w$ with $T(u)=u$ and $T(w)=-w$, namely, $u=(v+T(v))/2$ and $w=(v-T(v))/2$.