# 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

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 set of vectors whose infinity norm is a given constant, c, forms the surface of a hypercube with edge lengthÂ 2*c*.

where *D* is the domain of *f* (and the integral amounts to a sum if *D* is a discrete set).

For complex continuous functions over a compact space, this turns it into a C* algebra.