I have just discovered that for many years I have been using “can not” where “cannot” would be clearer. For example, concerning Riemann sums, I wrote
The Riemann lower sum for corresponding to a partition of can not be greater than the Riemann upper sum for corresponding to a partition of , even if and are different.
Concerning uniform convergence of sequences of real valued functions defined on a domain , I wrote
Let be a non-empty subset of and suppose that is a bounded function from to , i.e., is a bounded subset of . Let be a sequence of functions from to . Suppose that all of the functions are unbounded on . Then can not converge uniformly on to .
It looks like I haven’t used “can not” all that often in my second-year analysis notes. But it certainly never worried me when I did use it.
I had been ignoring the clue given me by Microsoft’s automatic spelling and grammar correction, which kept telling me to use “cannot” instead of “can not”. But recently this emerged as a difference of opinion with a colleague, and I am now convinced that “cannot” really is safer.
The difference is that “cannot” only means “is not able to”, while “can not” can have that meaning, but can also mean “is able not to”. Although I was aware of this second meaning, I was convinced that “can not” was common usage, and that it would be sufficiently rare to think of the other usage that I should not worry. However, various Google searches have convinced me that, at least in maths, I am in a relatively small minority with my use of “can not”. So I am going to change to “cannot” from now on.
But did McEnroe say “You can not be serious”, or did he say “You cannot be serious”? http://www.youtube.com/watch?v=ekQ_Ja02gTY