Limited illustration

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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

Schematic representation of a restricted (red) and an unlimited function (blue). The values ​​of the restricted function remain within the dashed lines over their entire domain. The values ​​of the unlimited function go towards infinity.

General is called a figure

limited when their image set is limited . More specifically, this means:

.
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 inclusions then apply

.

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