TITLE: What is the sense of the equality sign =?
QUESTION [7 upvotes]: Sometimes in real analysis, we write an equality and specify that is true in the sense of the $L^2$ norm.
My question is, when in mathematics we write '=' and don't specify anything, what we really mean?
Do we mean uniform convergence, more than that, or less than that?

REPLY [6 votes]: Building on Jesse Madnick's excellent answer, one way of interpreting

"$f = g$ in the sense of the $L^2$ norm"

is that what it really means is

"$\langle f \rangle = \langle g \rangle$, where $\langle f \rangle$ and $\langle g \rangle$ denote that $L^2$ equivalence classes of $f$ and $g$ respectively"

but we're being lazy in our notation and using just the plain function symbols $f$ and $g$ to stand for their equivalence classes, in order to avoid an excessive profusion of brackets.
Or, alternatively, we could interpret $f$ and $g$ as being the the equivalence classes, and say that we're being lazy by using them elsewhere as functions without explicitly picking a representative function from each class.
Either way, this habit of using the same symbol to denote both an equivalence class and a representative member of it is common in many areas of mathematics, since it makes the notation much less cluttered.  It does, however, mean that we sometimes need to explicitly clarify in which sense we are using a particular symbol.