direct sum of even/odd functions (example)

Example. Direct sum of even and odd functions

Let us define the sets

$F$$=$$\mathrm{\{}f|f\text{ is a function from}ℝ\text{ to}ℝ\},$

$F_{+}$$=$$\mathrm{\{}f\in F|f(x)=f(-x)\text{for all}x\in ℝ\},$

$F_{-}$$=$$\mathrm{\{}f\in F|f(x)=-f(-x)\text{for all}x\in ℝ\}.$

In other words, $F$ contain all functions from $ℝ$ to $ℝ$, $F_{+}\subset F$contain all even functions, and $F_{-}\subset F$ contain all odd functions.All of these spaces have a natural vector space structure:for functions $f$ and $g$ we define$f+g$ as the function $x\mapsto f(x)+g(x)$. Similarly, if $c$ isa real constant, then $cf$ is thefunction $x\mapsto cf(x)$. With these operations, the zero vectoris the mapping $x\mapsto 0$.

We claim that $F$ is the direct sum of $F_{+}$ and $F_{-}$, i.e.,that

$F$$=$$F_{+}\oplus F_{-}.$(1)

To prove this claim, let us first note that $F_{\pm }$ are vector subspaces of $F$.Second, given an arbitrary function $f$ in $F$, we can define

$f_{+}(x)$$=$${\displaystyle \frac{1}{2}}\left(f(x)+f(-x)\right),$

$f_{-}(x)$$=$${\displaystyle \frac{1}{2}}\left(f(x)-f(-x)\right).$

Now $f_{+}$ and $f_{-}$ are even and odd functions and $f=f_{+}+f_{-}$.Thus any function in $F$ can be split into two components $f_{+}$ and $f_{-}$,such that $f_{+}\in F_{+}$ and $f_{-}\in F_{-}$.To show that the sum is direct, suppose $f$ is an element in $F_{+}\cap F_{-}$.Then we have that $f(x)=-f(-x)=-f(x)$, so $f(x)=0$ for all $x$, i.e., $f$ isthe zero vector in $F$. We have established equation 1.