Supreme norm

The supremum norm of the real arctangent function is . Even if the function does not accept this value anywhere, it still forms the smallest upper bound. ${\displaystyle \pi /2}$

In mathematics, the supremum norm (also called infinity norm ) is a norm on the function space of limited functions . In the simplest case, a real- or complex- restricted function , the supremum is the upper bound of the sums of the function values. In a more general way one considers functions whose target set is a normalized space , and the supremum norm 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 in functional analysis when studying standardized spaces.

definition

Let be a non-empty set and a normalized space , then denotes the function space of the bounded functions from to . The supreme norm on this function space is then the mapping ${\displaystyle M}$ ${\displaystyle (Y,\|\cdot \|_{Y})}$ ${\displaystyle B(M,Y)}$ ${\displaystyle M}$ ${\displaystyle Y}$

${\displaystyle \|\cdot \|_{\infty }\colon B(M,Y)\rightarrow \mathbb {R} }$

With

${\displaystyle \|f\|_{\infty }:=\sup _{x\in M}\|f(x)\|_{Y}}$.

The supremum norm of a function is therefore the supremum of the norms of all function values ​​and thus a nonnegative real number. Here it is important that the function is limited, otherwise the supremum can become infinite. The room is also referred to as . ${\displaystyle B(M,Y)}$ ${\displaystyle \ell ^{\infty }(M,Y)}$

example

If one chooses the open unit interval as the set and the set of real numbers with the norm of absolute value as the target space , then the space of the bounded real-valued functions is on the unit interval and the supremum norm is through ${\displaystyle M=(0,1)}$ ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle |\cdot |}$ ${\displaystyle B(M,Y)}$

${\displaystyle \|f\|_{\infty }=\sup _{0<x<1}|f(x)|}$

given. For example, the supremum norm of the linear function is the same in this interval . The function does not take this value within the interval, but comes as close as desired. If you choose the closed unit interval instead , the value is accepted and the supremum norm corresponds to the maximum norm . ${\displaystyle f(x)=x}$ ${\displaystyle 1}$ ${\displaystyle M=[0,1]}$ ${\displaystyle 1}$

properties

Norm axioms

The supremum norm fulfills the three norm axioms definiteness , absolute homogeneity and subadditivity . The definiteness follows on from the definiteness of the norm over ${\displaystyle f\in B(M,Y)}$ ${\displaystyle \|\cdot \|_{Y}}$

${\displaystyle \|f\|_{\infty }=0\;\Leftrightarrow \;\sup _{x\in M}\|f(x)\|_{Y}=0\;\Rightarrow \;\forall x:\;\|f(x)\|_{Y}=0\;\Rightarrow \;\forall x:\;f(x)=0\;\Rightarrow \;f=0}$,

since if the supremum of a set of nonnegative real or complex numbers is zero, then all of those numbers must be zero. The absolute homogeneity follows for real or complex from the absolute homogeneity of the standard over ${\displaystyle \alpha }$ ${\displaystyle \|\cdot \|_{Y}}$

${\displaystyle \|\alpha f\|_{\infty }=\sup _{x\in M}\|\alpha f(x)\|_{Y}=\sup _{x\in M}|\alpha |\|f(x)\|_{Y}=|\alpha |\sup _{x\in M}\|f(x)\|_{Y}=|\alpha |\,\|f\|_{\infty }}$.

The subadditivity (or triangle inequality ) follows from the subadditivity of the norm over ${\displaystyle f,g\in B(M,Y)}$ ${\displaystyle \|\cdot \|_{Y}}$

${\displaystyle \|f+g\|_{\infty }=\sup _{x\in M}\|f(x)+g(x)\|_{Y}\leq \sup _{x\in M}(\|f(x)\|_{Y}+\|g(x)\|_{Y})\leq \sup _{x\in M}\|f(x)\|_{Y}+\sup _{x\in M}\|g(x)\|_{Y}=\|f\|_{\infty }+\|g\|_{\infty }}$,

where it was also used that the supremum of the sum of two functions is limited by the sum of the suprema, which can be seen by looking at the function values ​​point by point.

Other properties

• If the image space is complete , i.e. a Banach space , then it is also the entire functional space . ${\displaystyle B(M,Y)}$
• Is finite, then every function is bounded from to , so it is true . In particular , if one chooses for one , the natural identification of with gives a definition of the supremum norm on this Cartesian product . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle Y}$ ${\displaystyle B(M,Y)=\mathrm {Abb} (M,Y)}$ ${\displaystyle M=\{1,...,n\}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \mathrm {Abb} (M,Y)}$ ${\displaystyle Y^{n}}$
• In particular, one can consider the supremum norm in Euclidean space . In this case it is also referred to as the maximum norm . ${\displaystyle \mathbb {R} ^{n}}$
• If not finite or infinitely dimensional , then not every closed , bounded subset of is automatically compact . ${\displaystyle M}$ ${\displaystyle Y}$ ${\displaystyle B(M,Y)}$
• If it is not finite or infinitely dimensional, then it is not equivalent to all norms . ${\displaystyle M}$ ${\displaystyle Y}$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle B(M,Y)}$
• The supremum norm induces precisely the topology of uniform convergence on a space of limited functions .
• If the target area is or , then functions can not only be added point by point, but also multiplied. The supremum norm is then sub-multiplicative , that is . With pointwise multiplication, the space becomes a commutative Banach algebra . In the case this is even a C * -algebra . ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle Y=\mathbb {C} }$ ${\displaystyle B(M,Y)}$ ${\displaystyle \|f\cdot g\|_{\infty }\leq \|f\|_{\infty }\cdot \|g\|_{\infty }}$ ${\displaystyle B(M,Y)}$ ${\displaystyle Y=\mathbb {C} }$
• One can naturally generalize the notion of the restricted function and the supreme norm to vector bundles in which each fiber is a normalized space. The supremum norm is then a norm on the space of the restricted intersections of this vector bundle.

See also

• Essential Supremum

literature

• Dirk Werner : Functional Analysis . 6th, corrected edition, Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .

Individual evidence

1. ↑ Dirk Werner: Functional Analysis . 2005, p. 3 .