The Lecturer's Notation is Probably Good Enough
Date Written: June 14, 2024; Last Modified: June 14, 2024Sometimes, there will be a lecturer whose notation is particularly disagreeable. Perhaps they write \(\dot x(t)\) instead of \(x’(t)\) and you take issue with that. Maybe they use \(n\) to represent an arbitrary real number instead of your preferred letter \(z\) (equally disgusting, by the way). As tempting as it is to modify the notation to be more palatable in your notes, this is generally speaking a horrible idea, at least in my own experience.
First, keeping track of where your notation differs from the lecturer’s in real time costs precious attention, attention which one sometimes cannot afford to spend.
Second, it’s easy to have a lapse in focus and accidentally switch to the lecturer’s (worse) notation for just a line or two. A month and a half later, when you’re reviewing a difficult proof to prepare for an exam, you may be left wondering how an \(n\) turned into a \(z\) for no apparent reason. I know it’s happened to me.
As a less contrived example, I have seen \(C_c(X)\) and \(C_0(X)\) both be used to describe the space of compactly supported continuous real-valued functions on a space \(X\). However, I have also seen \(C_0(X)\) be used to denote the space of continuous real-valued functions on a space \(X\) that “vanish at \(\infty\)” (or, the closure of \(C_c(X)\) in \(C(X)\)). These conventions can change from lecturer to lecturer, and switching them around at the whim of personal preference is heavily prone to error.
Finally, as far as exams go, the notation on the exam is more likely to follow the lecturer’s dispicable notation than your pristine and aesthetically optimal notation. It is always a great shame when I see someone make mistakes due to incorrect notation alone.
That being said, some notational choices are purely cosmetic and determined by nothing more than tradition: for instance, using \(O(x)\), \(\lesssim x\), and \(\ll x\) are (most likely) the exact same thing.