Continuous function with compact carrier
A continuous function with a compact support is a special continuous function that only takes the value 0 outside of a compact unit . Such functions play an important role in functional analysis , as well as in stochastics and measure theory , where they are used as a separating family for sets of measures and the definition of convergence concepts.
definition
A topological space and a standardized space as well as a mapping are given
- .
The mapping is called a continuous function with compact support if the support of the function, i.e. the set
is a compact set and the map is continuous . It is therefore true that the archetypes of open sets (with regard to the topology generated by) are open again under , i.e. are contained in. If there is a metric space, this means that for all sequences that converge against , the image sequence converges against .
The set of all continuous functions with compact support is usually denoted by or . If it is clear which rooms are involved, you do not specify them; accordingly, the designations or are often found
structure
If one defines the addition and the scalar multiplication in point-wise, that is
- ,
such is a vector space.
Furthermore, every continuous function with a compact support is also a constrained function .
For example is a metric space, then there exists due to the continuity of each item on a so
If the carrier is now covered by the open sets , then due to the compactness there is a finite index set , so that the carrier is covered. Thus applies
- .
So is limited. is thus a subspace of , the space of limited mappings . For topological spaces one can generalize this argumentation with the help of a covering of the support with sets of the form .
Because of the limitation, the definition of the supremum norm is through
makes sense and makes a standardized space.
Superordinate structures
is a subspace of , the space of continuous, infinitely vanishing functions and bounded continuous functions , so the implications hold
- .
In addition, for a locally finite measure (or Borel measure ) on a Hausdorff space, every continuous function with a compact support can also be integrated, there
as a result of local finiteness. So in this case .
Subordinate structures
The test functions are an important subspace of the continuous functions with a compact carrier .
Important statements
According to Riesz-Markow's theorem, any positive linear form can be used in a locally compact Hausdorff space
represent as
- ,
where is a uniquely determined radon level . A linear form is called positive if it always follows.
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 .
- Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .