Pointwise bounded family - Imported Knowlpedia Content

A family $\mathcal{F}$ of functions $f:X\to\mathbb{R}$ is pointwise bounded if for every $x\in X$ the set of values $\{f(x): f\in\mathcal{F}\}$ is bounded in $\mathbb{R}$ , equivalently

\forall x\in X,\quad \sup_{f\in\mathcal{F}} |f(x)| < \infty.

Remarks

Pointwise boundedness is weaker than being uniformly bounded (which requires one bound to work for all $x$ at once). Together with equicontinuity hypotheses, it appears in compactness results for subsets of spaces of continuous functions such as Arzelà–Ascoli.

Examples

On $[0,1]$ , the family $\{f_n\}_{n\ge 1}$ with $f_n(x)=x^n$ is pointwise bounded since $0\le f_n(x)\le 1$ for all $x$ and $n$ .
On $[0,1]$ , the family $f_n(x)=n x$ is not pointwise bounded (for any fixed $x>0$ , the values $n x$ are unbounded as $n\to\infty$ ).