In class my professor introduced the following terminology:
Let $J$ be an arbitrary infinite set and let $f:J\to\mathbb{R}$ be a real-valued function. Then the symbol $\sum_{j\in J}f(j)$ is the series determined by $J$, with terms $f(j)$. We say that $\sum_{j\in J}f(j)$ converges to $S$ unconditionally if for every $\epsilon>0$, there is a finite subset $F_{0}\subset J$ such that if $F$ is any finite subset of $J$ that contains $F_{0}$, then
$\bigg|\sum_{j\in F}f(j)-S\bigg|<\epsilon$.
Does anyone know of any textbooks which discuss this topic? I haven't seen it in Rudin's Principles of Mathematical Analysis or in Royden and Fitzpatrick's Real Analysis. Any suggestions are appreciated!
It may not be easy to find! Most likely your professor will eventually get synchronized with the assigned textbook, clearing up your questions.
You might find material on this site sufficient to put this in perspective. For a similar definition and discussion see this answer to the MSE question,
$\qquad$ Can we add an uncountable number of positive elements, and can this sum be finite?
So just go with the flow and keep studying! Consider asking your professor if there is some advantage to defining a series in this non-conventional manner. I would also want to know if the definition means that $S$, when it exists, must be unique (the wording seems to suggests that that is the intention).