§ Why L2 needs a quotient upto almost everywhere
- We want a norm to have the property that if and only if .
- But in a function space, we can have nonzero functions taht have measure zero. eg. the function that is on and zero everywhere else.
- Thus, such functions are such that .
- To prevent this and to allow the L2 norm to really be a norm, we quotient by the closed subspace of functions such that .
- This has the side effect such that iff , or that functions agree almost everywhere.