Uniformly bounded family - Imported Knowlpedia Content

A family $\mathcal{F}$ of functions $f:X\to\mathbb{R}$ is uniformly bounded if there exists $M<\infty$ such that

|f(x)|\le M \quad \text{for all } f\in\mathcal{F}\text{ and all } x\in X.

Equivalently, if every $f\in\mathcal{F}$ is bounded, then $\sup_{f\in\mathcal{F}}\|f\|_\infty<\infty$ in terms of the supremum norm.

Remarks

Uniform boundedness implies pointwise boundedness and is a natural hypothesis when working with the uniform metric. It is also one of the typical ingredients in compactness criteria alongside equicontinuity.

Examples

On $\mathbb{R}$ , the family $f_n(x)=\sin(nx)$ is uniformly bounded with $M=1$ .
On $[0,1]$ , the family $f_n(x)=n x$ is not uniformly bounded.