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Uniform convergence in supremum norm

For bounded real-valued functions, uniform convergence is equivalent to convergence in the supremum norm.

Uniform convergence in supremum norm: Let EE be a set and let fn,f:E→Rf_n,f:E\to\mathbb{R} be bounded functions. Then fn→ff_n\to f on EE if and only if

∥fn−f∥∞⟶0,\|f_n-f\|_\infty \longrightarrow 0,

where ∥g∥∞=sup⁡x∈E∣g(x)∣\|g\|_\infty=\sup_{x\in E}|g(x)| is the .

Remarks

In particular, uniform convergence is exactly convergence in the metric induced by the supremum norm, which underlies the equipped with ∥⋅∥∞\|\cdot\|_\infty.