Uniform norm
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In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number
This norm is also called the supremum norm or the Chebyshev norm. If f is a continuous function on a closed interval, or more generally a compact set, then 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 = (x1,...,xn) in finite dimensional coordinate space, it takes the form
The reason for the subscript "∞" is that
where
where D is the domain of f (and the integral amounts to a sum if D is a discrete set).
The binary function
is then a metric on the space of all bounded functions on a particular domain. A sequence { fn : n = 1, 2, 3, ... } converges uniformly to a function f if and only if
For complex continuous functions over a compact space, this turns it into a C* algebra.
