§ Why L2 needs a quotient upto almost everywhere

• We want a norm to have the property that $|x| = 0$ if and only if $x = 0$.
• But in a function space, we can have nonzero functions taht have measure zero. eg. the function that is $1$ on $\mathbb Q$ and zero everywhere else.
• Thus, such functions are $f \neq 0$ such that $|f| = 0$.
• To prevent this and to allow the L2 norm to really be a norm, we quotient by the closed subspace of functions such that $|f| = 0$.
• This has the side effect such that $f = g$ iff $|(f - g)| = 0$, or that functions agree almost everywhere.