direct sum of even/odd functions (example)
Example. Direct sum of even and odd functions
Let us define the sets
In other words, contain all functions from to ,
contain all even functions, and contain all odd functions.
All of these spaces have a natural vector space structure
:
for functions and we define
as the function . Similarly, if is
a real constant, then is the
function . With these operations
, the zero vector
is the mapping .
We claim that is the direct sum of and , i.e., that
(1) |
To prove this claim, let us first note that are vector subspaces of . Second, given an arbitrary function in , we can define
Now and are even and odd functions and .
Thus any function in can be split into two components and ,
such that and .
To show that the sum is direct, suppose is an element in .
Then we have that , so for all , i.e., is
the zero vector in . We have established equation 1.
Title | direct sum of even/odd functions (example) |
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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 |