Another Notational Rant

This time, the physicists have nothing to do with it: real analysis has nobody else to blame.

Given a nonempty set X, and f₁, f₂, ..., f_n functions from X to , we define two functions and respectively , also from X to .

It is pretty obvious, just by looking at them, that the two functions are somehow related to each other (using terms very loosely here). Now if I tell you that one of the functions, applied to some x in X, gives

and the other one gives

is it obvious which one is which?

I am not sure how established the notation is, but I suspect it is not an uncommon thing. Admittedly, it is often difficult to think up clear notation, but when one can interchange the two without any obvious problems...that's not good.

What this is for: proving the Stone-Weierstrass Theorem. A very powerful theorem, by the way.

In contrast, as an example, the notation for upper and lower (Darboux) sums in discussing the definition of integrals is very, very clear. L(f,P) for the lower sum (of f with respect to a partition P), U(f,P) for the upper sum. What more do you want?

(Oh. I guess I should mention this. Apparently, the means "sup" or "join" and the means "inf" or "meet". Imagine that.)

As a reader may surmise, I am studying for the final exam. While the schedule this term actually is fairly decent (e.g. no two exams are on the same day), I am pretty much toast anyway. Not good to have the three most challenging exams in the same week. Not good at all.