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 length 2c.

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.