This excerpt from a journal paper says if X is a uniform space and $A$ is a subset of $X$, then $A$ is said to be bounded if for each entourage $V$, "there exists a finite set $F$ and a positive integer $m$ such that $A\subseteq V^m[F]$". My question is, what does $V^m[F]$ mean?
The $V^m$ part is clear enough; you just compose $V$ with itself $m$ times. But I haven't encountered a set in brackets after an entourage. If it said $V^m[x]$ where $x\in X$, that would make complete sense. How are the two notations related? If $F=\{x_1,...,x_n\}$, then does $V^m[F]=\cup_{i=1}^n V^m[x_i]$ or something?
If $R$ is a relation on $X$ (so $R \subseteq X \times X$) then $R[F]$ is just the "functional image" for a subset $F$ of $X$:
$$R[F]=\{y \in X: \exists x \in F: (x,y) \in R\}$$
So yes, this does commute with unions as you suggested. This is the same definition in essence as $f[F]$ for a function $f: X \to Y$ and $F \subseteq X$; note that a function is just a special case of a relation.