When = is not =

This morning I had a nice discussion on the NNTP group it.scienza.matematica about the use of $latex \arcsin$. In my opinion, we should abandon the old habit to define $latex \arcsin$ as the inverse function of some restriction of the sine function. I mean that it would be nice to write that the solution of the equation $latex \sin x = y$ is $latex x \in \arcsin y$. And this could probably prevent young students from manipulating goniometric formulas without thinking about it.

More generally, think of the Landau notation little-oh. We all know that, given a function $latex f$ and an accumulation point $latex p$ for the domain of $latex f$, the notation $latex o(f)$ means nothing but the set of all those functions $latex u$, defined on a neighborhood of $latex p$, such that

$latex \lim_{x \to p} \frac{u(x)}{f(x)}=0$.

Hence the formula $latex u=o(f)$ is a big mistake. We should always write $latex u \in o(f)$. Let me give an example that I found while acting as a referee. The authors had the assumption

$latex c = I(u)+o(1)$.

Then they wrote

$latex I(w) = \ldots = I(u)+o(1) = c$.

This is false, of course! They could prove that $latex I(w)=c+o(1)$, but they had no reason to simplify little-oh's like they were numbers!

Anyway, I understand that Landau's symbols are so useful and popular that I can't change its usage. So, be careful!