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
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)|
|Date of creation||2013-03-22 13:34:24|
|Last modified on||2013-03-22 13:34:24|
|Last modified by||mathcam (2727)|