Support (mathematics): Difference between revisions
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In [[mathematics]], the '''support''' of a function with values in a [[pointed set]], that is, a set with some distinguished element ω, is the set of values of the argument (a [[subset]] of the [[domain]]) for which the function does not take the value ω. | |||
In [[algebraic structure]]s with a [[zero element]], such as [[ring (mathematics)|rings]], the point of interest is usually the zero, so that the support of a function is the set of arguments where it takes a non-zero value. | |||
In [[analysis]], the definition of the support of a function ''f'' defined on a [[topological space]] with values in the real line is modified to denote the closure of the set on which ''f''(''x'') ≠ 0. | |||
An analogous concept can be introduced for [[distribution (mathematics)|distributions]] (generalized functions) or [[measure (mathematics)|measures]]. | An analogous concept can be introduced for [[distribution (mathematics)|distributions]] (generalized functions) or [[measure (mathematics)|measures]]. | ||
[[ | A function is said to be '''supported on''' a set ''S'' if its support is a [[subset]] of ''S''. | ||
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Latest revision as of 17:00, 23 October 2024
In mathematics, the support of a function with values in a pointed set, that is, a set with some distinguished element ω, is the set of values of the argument (a subset of the domain) for which the function does not take the value ω.
In algebraic structures with a zero element, such as rings, the point of interest is usually the zero, so that the support of a function is the set of arguments where it takes a non-zero value.
In analysis, the definition of the support of a function f defined on a topological space with values in the real line is modified to denote the closure of the set on which f(x) ≠ 0.
An analogous concept can be introduced for distributions (generalized functions) or measures.
A function is said to be supported on a set S if its support is a subset of S.