Uniform norm

The supremum (also called infinity norm) is in mathematics is a norm on the function space of bounded functions. In the simplest case of a real - or complex-valued bounded function the supremum norm is the supremum of the sums of the function values ​​. More generally, it functions, the target amount is a normed space, and the supremum is then the supremum of the norms of the function values ​​. For continuous functions on a compact set the maximum norm is an important special case of the supremum norm.

The supremum norm plays a central role, especially in the functional analysis in the study of normed spaces.

Definition

Let be a non-empty set and a normed space, then called by the function space of bounded functions. A function thus provides an element to a vector with finite norm. The supremum of this function space is then the mapping

With

The supremum norm of a function is therefore the supremum of the norms of all function values ​​and thus a non-negative real number. It is important that the function is bounded, because otherwise the supremum may be infinite. The space is also called.

Example

If you choose the amount of the open unit interval and as target space the set of real numbers with the standard amount, then the space of bounded real-valued functions on the unit interval and the supremum is by

Given. For instance, the supremum of the linear function in this interval is the same. The function takes this value, although within the interval not to, however, comes close to him any. If one chooses instead the closed unit interval, then the value is accepted and the supremum corresponds to the maximum norm.

Properties

Standard axioms

The supremum norm satisfies the three standard axioms definiteness, absolute homogeneity and subadditivity. The definiteness follows from the definiteness of the norm on

Because, if the supremum of a lot of non-negative real or complex numbers is zero, all of these numbers must be zero. The absolute homogeneity follows for real or complex from the absolute homogeneity of the standard on

The sub-additivity (or triangle ) follows from the subadditivity of the norm on

Was being used also that the supremum of the sum of two functions is limited by the sum of the Suprema, which is evident by pointwise analysis of the function values ​​.

Other properties

• Is the image space completely, ie a Banach space, so it is the entire function space.
• Is not finite, then is not every closed, bounded subset of automatically compact.
• Is not finite, it is not at all equivalent to standards.
• If the target space or, then can add functions in not only pointwise but also multiply. The supremum is then submultiplicative, that is,. The room is equipped with the pointwise multiplication to a commutative Banach algebra. In case this is even a C * - algebra.