Uniformly bounded family

Let $X$ be a set and let $\mathcal{F}\subseteq \mathbb{R}^X$ be a family of real-valued functions.

The family $\mathcal{F}$ is uniformly bounded if there exists $M<\infty$ such that $\forall f\in\mathcal{F}\ \forall x\in X:\ |f(x)|\le M.$

Equivalently, writing the sup norm $\|f\|_\infty=\sup_{x\in X}|f(x)|$ (possibly $+\infty$), uniform boundedness is: $\sup_{f\in\mathcal{F}} \|f\|_\infty <\infty.$

Uniform boundedness is stronger than pointwise boundedness ; the distinction is central in compactness criteria such as Arzelà–Ascoli .

Examples:

• On $X=[0,1]$, the family $\{f_n(x)=x^n\}$ is uniformly bounded by $1$.
• On $X=\mathbb{R}$, the family $\{f_n(x)=x/n\}$ is not uniformly bounded (the supremum over $x$ is infinite for each fixed $n$), even though it is pointwise bounded.