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[parent] direct sum of even/odd functions (example) (Example)

Example. Direct sum of even and odd functions

Let us define the sets

$\displaystyle F$ $\displaystyle =$ $\displaystyle \{ f\, \vert\, f\,$ is a function from$\displaystyle \, \mathbb{R}\,$    to$\displaystyle \, \mathbb{R}\},$  
$\displaystyle F_+$ $\displaystyle =$ $\displaystyle \{ f\in F \,\vert\, f(x)=f(-x) \,$for all$\displaystyle \, x\in \mathbb{R}\},$  
$\displaystyle F_-$ $\displaystyle =$ $\displaystyle \{ f\in F \,\vert\, f(x)=-f(-x)\,$for all$\displaystyle \, x\in \mathbb{R}\}.$  

In other words, $ F$ contain all functions from $ \mathbb{R}$ to $ \mathbb{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

$\displaystyle F$ $\displaystyle =$ $\displaystyle F_+ \oplus F_-.$ (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

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

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.



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See Also: direct sum of Hermitian and skew-Hermitian matrices, sum and product and quotient of functions


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Cross-references: equation, sum, components, vector subspaces, mapping, zero vector, operations, real, structure, vector space, odd functions, even functions, functions, contain, even and odd functions, direct sum
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This is version 3 of direct sum of even/odd functions (example), born on 2003-04-16, modified 2004-03-23.
Object id is 4191, canonical name is DirectSumOfEvenoddFunctionsExample.
Accessed 5539 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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