Section G.3 Variable hygiene
When doing analysis, you should in principle be asking yourself the following sorts of questions every time you write down a variable (e.g. \(\varepsilon,\delta,N,f,x\)):
- What does this variable depend on, or not depend on? Does it secretly depend on something I haven’t introduced yet? Will this be clear to my reader?
- Does the statement of the problem use the same symbol? If so, does it mean exactly the same thing here, or something else? If it means something else, will this be clear to my reader?
- Have I already used this symbol in my solution? If so, does it mean exactly the same thing here, or something else? If something else, will this be clear to my reader?
When introducing a variable for the first time, you similarly want to make its logical status clear. If you simply write ‘\(\varepsilon\gt0\)’, for instance, it may not be obvious whether you mean ‘Let \(\varepsilon\gt0\)’, ‘For all \(\varepsilon\gt 0\)’, ‘There exists \(\varepsilon \gt 0\)’ or something else.
Another thing to be careful about is the ‘type’ of the symbols you use, e.g.
- a real number,
- an element of some metric space,
- a subset of some metric space,
- a collection of subsets of a metric space,
- a function between metric spaces or
- a metric space of function spaces between metric spaces.
Hopefully this is clear to both you and your reader, and it also of course determines what sort of statements involving a symbol make sense. For instance,
- ‘\(3=\{x_1,x_2,x_3\}\)’ doesn’t make much sense, since the left hand side is a number while the right hand side is a set. Maybe the author is trying to tell us that the set \(\{x_1,x_2,x_3\}\) has three elements? Unclear.
- ‘\(\{1,2\} \in \{1,2,3\}\)’ isn’t true, since the set \(\{1,2\}\) is not an element of the set \(\{1,2,3\}\text{.}\) It is a subset, though, and so we can write \(\{1,2\} \subseteq \{1,2,3\}\text{.}\)
- ‘The closure \(\overline 1\) of \(1\)’ doesn’t make much sense, since \(1\) is a number, while closure is an operation defined on sets. Perhaps the author means \(\overline{\{1\}}\text{?}\)
- If \(f \maps [a,b] \to \R\text{,}\) then expressions like \(\n{f(x)-f(y)}_{C^0([a,b])}\) are deeply ambiguous. Does the author mean \(\sup_{x \in [a,b]} \abs{f(x)-f(y)}\text{,}\) or \(\sup_{y \in [a,b]} \abs{f(x)-f(y)}\text{,}\) or \(\sup_{x,y \in [a,b]} \abs{f(x)-f(y)}\text{?}\) Or maybe they mean something a simple as \(\abs{f(x)-f(y)}\text{?}\)
