equibounded

Let $X$ and $Y$ be metric spaces. A family $F$ of functions from $X$ to $Y$ is said to be equibounded if there exists a bounded subset $B$ of $Y$ such that for all $f\in F$ and all $x\in X$ it holds $f(x)\in B$.

Notice that if $F\subset {𝒞}_b(X,Y)$ (continuous bounded functions) then $F$ is equibounded if and only if $F$ is bounded (with respect to the metric of uniform convergence).