direct sum of even/odd functions (example)

Example. Direct sumMathworldPlanetmathPlanetmathPlanetmath of even and odd functions

Let us define the sets

F = {f|f is a function from to},
F+ = {fF|f(x)=f(-x)for allx},
F- = {fF|f(x)=-f(-x)for allx}.

In other words, F contain all functions from to , F+F contain all even functions, and F-F contain all odd functions. All of these spaces have a natural vector spaceMathworldPlanetmath structureMathworldPlanetmath: for functions f and g we define f+g as the function xf(x)+g(x). Similarly, if c is a real constant, then cf is the function xcf(x). With these operationsMathworldPlanetmath, the zero vector is the mapping x0.

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

F = F+F-. (1)

To prove this claim, let us first note that F± are vector subspaces of F. Second, given an arbitrary function f in F, we can define

f+(x) = 12(f(x)+f(-x)),
f-(x) = 12(f(x)-f(-x)).

Now f+ and f- are even and odd functions and f=f++f-. Thus any function in F can be split into two componentsMathworldPlanetmathPlanetmath f+ and f-, such that f+F+ and f-F-. To show that the sum is direct, suppose f is an element in F+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