Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns, to real- or complex-valued bounded functions ⁠

⁠, 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 ⁠

⁠ under the metric derived from the uniform norm if and only 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 finite dimensional coordinate space, it takes the form: This is called the

Uniform norms are defined, in general, for bounded functions valued in a normed space.

, there is an extended norm defined by This is in general an extended norm since the function

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

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.

The uniform metric between two bounded functions

is defined by The uniform metric is also called the Chebyshev metric, after Pafnuty Chebyshev, who was first to systematically study it.

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.

converges uniformly to a function

induced by the uniform extended norm is the same as the uniform extended metric on

is said to converge uniformly to a function

form a fundamental system of entourages of a uniformity on

The uniform convergence is precisely the convergence under its uniform topology.

is a metric space, then it is by default equipped with the metric uniformity.

The set of vectors whose infinity norm is a given constant,

forms the surface of a hypercube with edge length

; the integral amounts to a sum if

is a discrete set (see p-norm).