Uniform norm

In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions f defined on a set S the non-negative number This norm is also called the supremum norm, the Chebyshev norm, or the infinity norm. The name 'uniform norm' derives from the fact that a sequence of functions { f n } {displaystyle {f_{n}}} converges to f under the metric derived from the uniform norm if and only if f n {displaystyle f_{n}} converges to f {displaystyle f} uniformly. 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. If f is a continuous function on a closed 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, for the case of a vector x = ( x 1 , … , x n ) {displaystyle x=(x_{1},dots ,x_{n})} in finite dimensional coordinate space, it takes the form The reason for the subscript '∞' is that whenever f is continuous