In mathematical analysis, the uniform norm (or sup norm) assigns, to real- or complex-valued bounded functions ⁠${\displaystyle f}$⁠ defined on a set ⁠${\displaystyle S}$⁠, the non-negative number

This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. The name "uniform norm" derives from the fact that a sequence of functions ⁠${\displaystyle \left\{f_{n}\right\}}$⁠ converges to ⁠${\displaystyle f}$⁠ under the metric derived from the uniform norm if and only if ⁠${\displaystyle f_{n}}$⁠ converges to ⁠${\displaystyle f}$⁠ uniformly.

If ⁠${\displaystyle f}$⁠ is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if ⁠${\displaystyle x}$⁠ is some vector such that ${\displaystyle x=\left(x_{1},x_{2},\ldots ,x_{n}\right)}$ in finite dimensional coordinate space, it takes the form:

This is called the ${\displaystyle \ell ^{\infty }}$-norm.

Uniform norms are defined, in general, for bounded functions valued in a normed space. Let ${\displaystyle X}$ be a set and let ${\displaystyle (Y,\|\|_{Y})}$ be a normed space. On the set ${\displaystyle Y^{X}}$ of functions from ${\displaystyle X}$ to ${\displaystyle Y}$, there is an extended norm defined by

This is in general an extended norm since the function ${\displaystyle f}$ may not be bounded. Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the uniform norm on ${\displaystyle Y^{X}}$. Note that the definition of uniform norm does not rely on any additional structure on the set ${\displaystyle X}$, although in practice ${\displaystyle X}$ is often at least a topological space.

The convergence on ${\displaystyle Y^{X}}$ in the topology induced by the uniform extended norm is the uniform convergence, for sequences, and also for nets and filters on ${\displaystyle Y^{X}}$.

We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called uniformly closed and closures uniform closures. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on ${\displaystyle A.}$ For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on ${\displaystyle [a,b]}$ is the uniform closure of the set of polynomials on ${\displaystyle [a,b].}$