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Uniform norm
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== Definition == Uniform norms are defined, in general, for [[bounded function]]s valued in a [[normed space]]. Let <math>X</math> be a set and let <math>(Y,\|\|_Y)</math> be a [[normed space]]. On the set <math>Y^X</math> of functions from <math>X</math> to <math>Y</math>, there is an [[extended norm]] defined by :<math>\|f\|=\sup_{x\in X}\|f(x)\|_Y\in[0,\infty].</math> This is in general an extended norm since the function <math>f</math> 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 <math>Y^X</math>. Note that the definition of uniform norm does not rely on any additional structure on the set <math>X</math>, although in practice <math>X</math> is often at least a [[topological space]]. The convergence on <math>Y^X</math> in the topology induced by the uniform extended norm is the [[uniform convergence]], for sequences, and also for [[net (mathematics)|nets]] and [[filter (mathematics)|filters]] on <math>Y^X</math>. 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 <math>A.</math> For instance, one restatement of the [[Stone–Weierstrass theorem]] is that the set of all continuous functions on <math>[a,b]</math> is the uniform closure of the set of polynomials on <math>[a, b].</math> For complex [[Continuous function (topology)|continuous]] functions over a compact space, this turns it into a [[C-star algebra|C* algebra]] (cf. [[Gelfand representation]]).
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