This article is about the function space norm. For the finite-dimensional vector space distance, see Chebyshev distance. For the uniformity norm in additive combinatorics, see Gowers norm.
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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]
The metric generated by this norm is called the Chebyshev metric, after Pafnuty Chebyshev, who was first to systematically study it.
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 binary function
is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence converges uniformly to a function if and only if
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.
Properties
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).
This article uses material from the Wikipedia article Infinity_norm, and is written by contributors.
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