[[Function space]]
# Uniform norm
**Uniform norm**[^aka] is a [[Normed vector space|norm]] on a [[Function space]] defined as the [[Poset#^sup|supremum]] of a function on its domain. #m/def/topology 
For the space $C(I)$, the uniform norm of a function $f : I \to \mathbb{C}$
$$
\begin{align*}
\|f\|_{\infty} = \sup \{ \abs{f(x)} : x \in I \}
\end{align*}
$$
The metric induces is the [[Supremum metric]], which is used to define [[Uniform convergence]]. 
As suggested by the notation, the uniform norm is a limit of the [[Lebesgue space|$p$-norm]] as $p \to \infty$.

[^aka]: Also called sup norm, Chebyshev norm, or infinity norm

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