Orthogonal Complement of Even Functions.
August 5, 2012
Question: Consider the subspace of consisting of even functions (that is, functions with ). Find the orthogonal complement of .
One Solution. It’s easy to prove is a subspace. Then, there is a representation of any function in this space by adding odd and even functions together; more precisely, given we have that is even and is odd and . For uniqueness, note that if , then for each , giving us that . Hence, the orthogonal complement of is the set of odd functions.
Here’s another solution that "gets your hands dirty" by manipulating the integral.
Another Solution. We want to find all such that for every even function . This is equivalent to wanting to find all such with . Assume is in the orthogonal complement. That is,
The last equality here re-parameterizes the first integral by letting , but note that our new gives us the negative sign.
We may choose since this is an even function, and we note that this gives us
Since , it must be the case that . [Note: The fact that this is only true "almost everywhere" is implicit in the definition of .] Hence, , giving us that .
We now have one direction: that if is in the orthogonal complement, then it will be odd. Now we need to show that if is any odd function, it is in the orthogonal complement. To this end, suppose is an odd function. Then by the above, we have
where the last equality comes from the fact that is odd.