Is $\sum a$ a customary (standard) shorthand for $\sum_{i\in\operatorname{dom}a} a_i$, where $a$ is an indexed family of say integers?
Sum without an index
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1http://math.stackexchange.com/questions/124322/a-cont$r$adiction-in-notation#comment287236_124322 – 2012-03-27
3 Answers
You will sometimes see it used that way, but in my view it’s a dismally poor abuse of notation. At the very least the index should appear somewhere in the expression: $\sum_ia_i$ is fine, given a reasonable context, or even $\sum a_i$, but $\sum a$ is at best annoying and at worst confusing, especially since $\sum_{k=1}^na$ has the completely different unambiguous meaning $na$.
Added: It occurs to me belatedly that there is one context in which I would not at all object to the notation $\sum a$: if $a$ is a finite set of real numbers, say, $\sum a$ is perfectly acceptable shorthand for $\sum\{x:x\in a\}$, just as in set theory $\bigcup a$ is unambiguously $\bigcup\{x:x\in a\}$ if $a$ is a set of sets.
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0Of course, when $a$ is e.g.$a$vector then it's not clear. Even more so as in that case $\sum_i a_i$ is suspicious, because basis-dependent. But for$a$sequence, not treated as an element of$a$vector space, I'd consider it perfectly obvious — No, the notation is not a programming language. Alas! — @mhum: actually, I use Einstein convention very often. I can't say I like it too much (the distinction between greek and roman letters in some contexts), but I certainly don't dislike it. In that convention, index variables always turn up two times, so you don't get what I criticised about $\sum a_i$. – 2012-03-28
Yes. IMO there's not much of a problem with it: in a Haskell-ish pseudo-lambda-calculus-notation $\begin{align} &\Sigma\ ::\ (J\text{ countable}, S\text{ additive})\Rightarrow\ (J \to S) \to S \\ &\Sigma f = \underbrace{f(j_1) + f(j_2) + \ldots}_{\text{all }j_k\in A} \end{align}$ with the more common general notation just being shorthand $ \sum_{i\in I}a_i := \Sigma\bigl(\lambda i.\ a_i\,\chi_I(i)\bigr) $ where $\chi_I(i)=1$ for $i\in I$ and $0$ otherwise.
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0The ambiguity also comes up with summation (see Brian M. Scott's edit), but as a set is never a function it can always be resolved. Was it this you meant, or the characteristic function $\chi$? – 2012-03-28
Often, yes. The $a_i$ need not be integers, and the index set can also be different from integers - it's usually understood from context. The same goes for products, $\prod a_i$.