Uniformly bounded family

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

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

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.