# direct sum of even/odd functions (example)

Example. Direct sum^{} of even and odd functions

Let us define the sets

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

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

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

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

$F$ | $=$ | ${F}_{+}\oplus {F}_{-}.$ | (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

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

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

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 |