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Concept
Support (mathematics)
Summary
In mathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of is a topological space, then the support of is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.
Suppose that is a real-valued function whose domain is an arbitrary set The of written is the set of points in where is non-zero:
The support of is the smallest subset of with the property that is zero on the subset's complement. If for all but a finite number of points then is said to have .
If the set has an additional structure (for example, a topology), then the support of is defined in an analogous way as the smallest subset of of an appropriate type such that vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than and to other objects, such as measures or distributions.
The most common situation occurs when is a topological space (such as the real line or -dimensional Euclidean space) and is a continuous real (or complex)-valued function. In this case, the of , , or the of , is defined topologically as the closure (taken in ) of the subset of where is non-zero that is,
Since the intersection of closed sets is closed, is the intersection of all closed sets that contain the set-theoretic support of
For example, if is the function defined by
then , the support of , or the closed support of , is the closed interval since is non-zero on the open interval and the closure of this set is
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that (or ) be continuous.
Functions with on a topological space are those whose closed support is a compact subset of If is the real line, or -dimensional Euclidean space, then a function has compact support if and only if it has , since a subset of is compact if and only if it is closed and bounded.
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