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 $c$ is a real constant, then $cf$ is the function $x\mapsto cf(x)$. With these operations, the zero vector is 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., $f$ is the zero vector in $F$. We have established equation 1.

Title	direct sum of even/odd functions (example)
Canonical name	DirectSumOfEvenoddFunctionsexample
Date of creation	2013-03-22 13:34:24
Last modified on	2013-03-22 13:34:24
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Example
Classification	msc 26A06
Related topic	DirectSumOfHermitianAndSkewHermitianMatrices
Related topic	ProductAndQuotientOfFunctionsSum