# Is the “$\mapsto$” notation considered “proper” mathematics?

A few years ago in an analysis class, I had a teacher tell me that the notation "$\mapsto$" (e.g. $f \mapsto f'$) wasn't "proper" mathematics. Since then, I have ardently avoided it, even to the point that if I saw it in a book, I found the book less credible. However, I recently discovered that I enjoy algebra much more than I ever enjoyed analysis and from the books I've recently devoured, I've noticed a certain ubiquity concerning the "$\mapsto$" notation in algebra. I'm currently reading Bhattacharya and he, in particular, makes regular use of it. This notation does seem to have some advantages. So basically, was my analysis professor just presenting his own aesthetic views as gospel or is this particular notation frowned upon in analysis while being embraced elsewhere?

• This notation is quite standard. And, to me, very visually pleasing. – Joe Johnson 126 Dec 2 '16 at 17:53
• It's just another way of defining a function/map. If I write $f(x) = x^2$ I can also write $x \mapsto x^2$. I see nothing wrong with it, not sure what your teacher was on. – ÍgjøgnumMeg Dec 2 '16 at 17:54
• Your teacher was wrong. That symbol just tells what the typical rule of assignment is without having to explicitly name the function. It's usually used along with a description of the domain and range. For example, one may speak of the map $\mathbb R\to \mathbb R$ given by $x\mapsto e^x$. This is the exponential function, of course, but notice that I haven't named the function explicitly. – MPW Dec 2 '16 at 17:55
• First of all, it's just a notation. Secondly, there is literally a branch of mathematics motivated by the notation: en.wikipedia.org/wiki/Lambda_calculus. So although one could have reservations about some notation, to call it "improper" seems strange, to say the least. – Pavel Čoupek Dec 2 '16 at 17:57
• @thelaughingalgebraist This notation is totally fine. – rschwieb Dec 2 '16 at 17:58

I have never heard of $\text{“}\mapsto\text{''}$ being considered improper mathematics. It can certainly be used in a slightly inexact way. For example, describing a map from naturals to naturals as $$\text{“x\mapsto 2x''}$$ rather than $$\text{“\mathbb{N}\rightarrow\mathbb{N}: x\mapsto 2x''}$$ is technically ambiguous, since we can't tell that this doesn't refer to a map $\mathbb{R}\rightarrow\mathbb{R}$. But this is a silly objection, since the context $\mathbb{N}$ would have been specified earlier in the text. Besides, far worse abuses of notation and general ambiguities are prevalent in math; there's no reason to single out $\text{“}\mapsto\text{''}$.

Having brought up ambiguity as a possible criticism, let me now defend it!

When writing informally (which is important! nobody likes just reading a completely formal proof), it's sometimes useful to refer to a vague thing before precisely defining it. E.g. $$\text{“Whenever [thing] is a naturally occuring [stuff], the map f\mapsto f' is [meh].''}$$ This kind of writing is horribly imprecise, yet can be very useful for describing the overall picture of a proof or construction, or for motivating a theorem. Math is a social activity, and even if we want to criticize $\text{“}\mapsto\text{''}$ as imprecise (which it really isn't, see above), that doesn't mean it's not "proper mathematics."

Bottom line:

• In the course of a rigorous proof, the onus is on the writer to make things completely precise. You can use whatever symbols you want, but you have to use them unambiguously.

• In the course of explaining something, there's only one rule: does what you write make things clearer? If you can explain something better by using the LaTeX symbol for Bart Simpson's face, great! (Well, fine - reactions may vary to that one. But my overall point stands, and in fact I'd say even this is okay if you don't mind people Groening at it.)

• is technically ambiguous Maybe that was teacher's point. I notice OP's example is $f \mapsto f'\,:$ if that notation was used in a context where function spaces and differential operators were not (yet) formally covered, it would have been a valid objection. – dxiv Dec 2 '16 at 18:02
• @dxiv Only if the context hadn't been fixed clearly in the text before that. Besides, that's not really a problem with the symbol "$\mapsto$," but rather of mathematical writing in general. – Noah Schweber Dec 2 '16 at 18:02
• Ok that makes me feel much better about it. I guess, like anything else, it is dependent on context. I feel like an idiot for having looked down on texts that used this notation for the past two years but it is nice to be able to let go of such silliness. Thank you for clarifying this. – TheLaughingAlgebrist Dec 2 '16 at 18:12

$\mapsto$ "mapsto" is the usual symbol for "mapsto" in "proper" mathematics.