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In mathematical analysis, the **uniform norm** (or **sup norm**) assigns to real- or complex-valued bounded functions defined on a set 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 converges to under the metric derived from the uniform norm if and only if converges to uniformly.^{[1]}

If 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 is some vector such that in finite dimensional coordinate space, it takes the form:

This is called the -norm.

Uniform norms are defined, in general, for bounded functions valued in a normed space. Let be a set and let be a normed space. On the set of functions from to , there is an extended norm defined by

This is in general an extended norm since the function 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 . Note that the definition of uniform norm does not rely on any additional structure on the set , although in practice is often at least a topological space.

The convergence on in the topology induced by the uniform extended norm is the uniform convergence, for sequences, and also for nets and filters on .

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 For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on is the uniform closure of the set of polynomials on

For complex continuous functions over a compact space, this turns it into a C* algebra (cf. Gelfand representation).

The **uniform metric** between two bounded functions from a set to a metric space is defined by

The uniform metric is also called the **Chebyshev metric**, after Pafnuty Chebyshev, who was first to systematically study it. In this case, is bounded precisely if is finite for some constant function . If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question; the convergence is then still the uniform convergence. In particular, a sequence converges uniformly to a function if and only if

If is a normed space, then it is a metric space in a natural way. The extended metric on induced by the uniform extended norm is the same as the uniform extended metric

on

Let be a set and let be a uniform space. A sequence of functions from to is said to converge uniformly to a function if for each entourage there is a natural number such that, belongs to whenever and . Similarly for a net. This is a convergence in a topology on . In fact, the sets

where runs through entourages of form a fundamental system of entourages of a uniformity on , called the **uniformity of uniform convergence** on . The uniform convergence is precisely the convergence under its uniform topology.

If is a metric space, then it is by default equipped with the metric uniformity. The metric uniformity on with respect to the uniform extended metric is then the uniformity of uniform convergence on .

The set of vectors whose infinity norm is a given constant, forms the surface of a hypercube with edge length

The reason for the subscript “ ” is that whenever is continuous and for some , then
where
where is the domain of ; the integral amounts to a sum if is a discrete set (see *p*-norm).

- L-infinity – Space of bounded sequences
- Uniform continuity – Uniform restraint of the change in functions
- Uniform space – Topological space with a notion of uniform properties
- Chebyshev distance – Mathematical metric