Definition talk:Pointwise Operation on Extended Real-Valued Functions

On this and the other matching page: I'm not too happy about the notation being used here, it seems a bit clumsy.

Surely elements of $\R$ are operated on by well-known operations: $+$, $\times$ etc. As $f(x)$ and $f(y)$ are actual elements of $\R$ then it makes sense to write $f(x) + f(y)$ (and so on) without the subscript on it, which will serve only to confuse. The question will arise "So what makes $+_{\overline \R}$ different from plain ordinary $+$?" and the answer will be: absolutely nothing.

If you want to distinguish between the operations on the functions and the operations on the values of the functions (which, once you have introduced the concept, there is rarely any reason to do so), then I suggest it might be better to modify the former, and write something like: $(f +' g) (x) := f(x) + g(x)$.

The other thing I'm not sure about is having a completely different page for this from the real-valued functions one. I agree that it's worth putting a full definition page in for real-valued functions because that's a basic entry point in analysis, but it feels like overkill to do the same thing for extended rvfs, on the grounds that if you're at this point in your studies, you will already have taken this concept on in the context of real-valued function, so don't need the whole thing explained blow-by-blow in this context as well.

In fact, the only main reason for having a separate page, i.e. that of the notation, vanishes if the first point above is actioned.

Thoughts? --prime mover 08:03, 7 April 2012 (EDT)

The problem with merging is that $\overline{\R}$ misses a critical structure: it is not a group (it is void to consider $-\infty + +\infty$). Therefore, many of the proofs need to be done specifically for $\overline{\R}$ (or arguments need to use only something like 'commutative semigroup', which $\overline{\R}$ actually is (wrt $+$). Point about the notation agreed; it was a rushed decision to distinguish, and may indeed very well lead to confusion. --Lord_Farin 08:13, 7 April 2012 (EDT)

The only further specialized page I have in mind is for vector spaces, because operations on operators are paramount in functional analysis (you probably recall yourself asking questions about those things). I still feel that this separate page should stay, on the grounds mentioned above. $\overline{\R}$ just behaves a bit strange sometimes and I want to explicate its intricacies as much as I can. --Lord_Farin 08:13, 7 April 2012 (EDT)

Fair enough - the usual rule applies: get the information in, and we can rearrange it as we need to in due course. --prime mover 08:19, 7 April 2012 (EDT)