In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number

This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . 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.

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 . In particular, if in finite dimensional coordinate space, it takes the form:

This is called the \ell^\infty-norm.

Definition

Uniform norms are defined, in general, for bounded functions valued in a normed space be a normed space, there is an extended norm defined by

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 is often at least a topological space.

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

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 For instance, one restatement of the Stone–Weierstrass theorem

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

Weaker structures inducing the topology of uniform convergence

Uniform metric

The uniform metric to a metric space is defined by

The uniform metric is also called the , after Pafnuty Chebyshev 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

is a normed space, then it is a metric space induced by the uniform extended norm is the same as the uniform extended metric

Uniformity of uniform convergence

be a uniform space. In fact, the sets

, called the uniformity of uniform convergence. 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.

Properties

forms the surface of a hypercube with edge length

is a discrete set (see p-norm).

See also

References