Limited illustration
As a limited figure or a limited function is referred to in the analysis and the functional analysis a picture whose image set is limited. Bounded mappings form a normalized vector space and contain many other important sets of mappings such as the continuous functions with compact support or the bounded continuous functions .
The concept of restricted mapping is to be distinguished from that of restricted linear mapping . For this class of images, only the image of restricted subsets is again restricted.
definition
General is called a figure
limited when their image set is limited . More specifically, this means:
- If a real-valued function or a complex-valued function , then this corresponds
- .
- The image set of the function is then clearly contained in a finite interval in the real-valued case or in a circle lying in the complex plane in the complex-valued case.
- If a standardized room has a norm, this corresponds
- .
- Is a metric space and so this corresponds
- .
In particular, no requirements are placed on the structure of the definition set.
The set of all restricted images from to is denoted by or by , if or or if is evident from the context.
Examples
Constrained sequences are constrained functions of, for example, or a general metric space.
The sine function is limited because it applies to all .
If a function is continuous, it is also bounded. Because as a constant function on the compact version assumes a maximum and a minimum and it applies .
The previous example is a special case of the following fact: If a compact topological space and a metric space, then every continuous mapping is bounded. Because of the continuity exists at every point a the inclusion, so that
applies. Due to the compactness of, however , the open cover defined in this way has a finite partial cover with and thus follows
- .
So is limited.
An example of a discontinuous bounded function is the Dirichlet function .
structure
If the structure of a vector space bears , then the addition and the scalar multiplication can be defined point by point,
- and ,
whereby the set of bounded maps naturally becomes a vector space.
If a normalized space is, a norm can be explained by
- ,
where the norm refers to. This is exactly the supremum norm , it is accordingly also referred to as or when all the spaces involved are clear.
If a Banach space is also complete , then it is also a Banach space.
If a space is compact , every continuous mapping is limited. Inclusion then applies
- .
If compact and a Banach space, then the continuous functions form a closed subspace of the bounded functions.
Important subspaces of the restricted mappings with values in are
- the continuous functions with a compact carrier ,
- the continuous functions that vanish at infinity and
- the bounded continuous functions .
The inclusions then apply
- .
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 .