Limited continuous function

The bounded continuous functions are a class of functions that have diverse applications in functional analysis or measure theory . For example, they appear as a separating family of finite measures on Borel's σ-algebra of a metric space, where they are used to define the weak convergence of measures . They are also used, for example, in the Riesz-Markow representation set .

definition

A topological space and a metric space are given . Then a function is called ${\displaystyle (X,\tau )}$ ${\displaystyle (S,d)}$

${\displaystyle f\colon X\to S}$

a bounded continuous function if its image is bounded, so

${\displaystyle \operatorname {diam} (f(X))=\sup\{d(f(x),f(y))\;|\;x,y\in X\}<\infty }$

holds, and it is continuous , i.e. archetypes of open sets (with regard to the topology generated by) are open again, i.e. are contained in. ${\displaystyle d}$ ${\displaystyle \tau }$

If stronger structures are defined on the definition and image set (for example a metric or standardized space as a definition set or a standardized space as an image set), the definitions of continuity and limitation are adapted accordingly.

The set of all continuous, bounded functions is denoted by or simply by if or if all spaces involved are clear. ${\displaystyle C_{b}(X,S)}$ ${\displaystyle C_{b}(X)}$ ${\displaystyle S=\mathbb {K} }$ ${\displaystyle C_{b}}$

structure

If a vector space , the addition and the scalar multiplication can be explained pointwise as ${\displaystyle S}$ ${\displaystyle C_{b}(X,S)}$

${\displaystyle (f+g)(x):=f(x)+g(x){\text{ sowie }}(\lambda f)(x):=\lambda f(x){\text{ für alle }}x\in X}$.

This means that there is also a vector space. If there is also a norm , i.e. a standardized space , then the supremum norm can be used ${\displaystyle C_{b}(X,S)}$ ${\displaystyle S}$ ${\displaystyle \|\cdot \|_{S}}$ ${\displaystyle C_{b}(X,S)}$

${\displaystyle \|f\|_{\operatorname {sup} }:=\sup _{x\in X}\|f(x)\|_{S}}$

provided, since all functions are restricted and the norm is thus well-defined.

The bounded, continuous functions are a subspace of the bounded functions and contain, as important subspaces, the -functions (the continuous functions that vanish at infinity), the continuous functions with compact support, and the test functions . ${\displaystyle C_{0}}$

literature

• Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
• Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .