# Carrier (mathematics)

In the mathematics of the designated carrier (Engl. Support ) usually the closure of the non-zero set of a function or other objects.

## Analysis

### Bearer of a function

The bearer of is usually referred to as or . ${\ displaystyle f}$${\ displaystyle \ operatorname {Tr} (f)}$${\ displaystyle \ operatorname {supp} (f)}$

Be a topological space and a function. The carrier of then consists of the closed envelope of the non-zero set of , formally: ${\ displaystyle A}$${\ displaystyle f \ colon A \ to \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle f}$

${\ displaystyle \ operatorname {Tr} (f) = \ operatorname {supp} (f): = {\ overline {\ {x \ in A \ mid f (x) \ neq 0 \}}}}$

### Carrier of a distribution

Be an open subset of and a distribution. One says that a point belongs to the bearer of , and writes if a function exists for every open environment of . ${\ displaystyle \ Omega}$${\ displaystyle \ mathbb {R} ^ {d}}$${\ displaystyle T \ in {\ mathcal {D}} '(\ Omega)}$${\ displaystyle x_ {0} \ in \ Omega}$${\ displaystyle T}$${\ displaystyle x_ {0} \ in \ mathrm {supp} (T)}$${\ displaystyle U \ subset \ Omega}$${\ displaystyle x_ {0}}$${\ displaystyle \ phi \ in {\ mathcal {D}} (U)}$${\ displaystyle \; T (\ phi) \ neq 0}$

If there is a regular distribution with continuous f , then this definition is equivalent to the definition of the carrier of a function (the function f ). ${\ displaystyle T}$${\ displaystyle T = T_ {f}}$

### Examples

Is with , then is , for the non-zero set of is whose closure is integer . The same is true for any polynomial function except the null function . ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle f (x) = x}$${\ displaystyle \ operatorname {supp} (f) = \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle \ mathbb {R} \ setminus \ left \ {0 \ right \}}$${\ displaystyle \ mathbb {R}}$

Is with , if , else , then is the amount . ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle f (x) = 1}$${\ displaystyle \ left | x \ right | <1}$${\ displaystyle 0}$${\ displaystyle \ operatorname {supp} (f)}$${\ displaystyle \ left \ {x: \ left | x \ right | \ leq 1 \ right \}}$

Is the characteristic function of , if , and , if , then is the carrier , i.e. the closure of . ${\ displaystyle \ chi _ {\ mathbb {Q}}}$${\ displaystyle \ mathbb {Q}: \ chi _ {\ mathbb {Q}} (x) = 1}$${\ displaystyle x \ in \ mathbb {Q}}$${\ displaystyle \ chi _ {\ mathbb {Q}} (x) = 0}$${\ displaystyle x \ in \ mathbb {R} \ setminus \ mathbb {Q}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {Q}}$

Be an open subset of the . The set of all continuous functions from to with compact support forms a vector space , which is denoted by. ${\ displaystyle U}$${\ displaystyle \ mathbb {R} ^ {d}}$${\ displaystyle U}$${\ displaystyle \ mathbb {R}}$${\ displaystyle C_ {c} (U)}$

The set of all smooth (infinitely often continuously differentiable) functions with compact support plays a major role in the theory of distributions as the set of “test functions” . ${\ displaystyle C_ {c} ^ {\ infty} (U)}$${\ displaystyle U}$

The delta distribution has the carrier , because the following applies: if it is over , then it is . ${\ displaystyle \ delta (f): = f (0)}$${\ displaystyle \ left \ {0 \ right \}}$${\ displaystyle \ omega: = \ mathbb {R} ^ {d} \ setminus \ left \ {0 \ right \}}$${\ displaystyle f}$${\ displaystyle C_ {c} ^ {\ infty} (\ omega)}$${\ displaystyle \ delta (f) = 0}$

## Sheaf theory

Let it be a sheaf of Abelian groups over a topological space . ${\ displaystyle F}$ ${\ displaystyle X}$

### Carrier of a cut

For an open subset and a section , the end of the set of those points for which the image of in the stalk is not equal to zero is called the carrier of , usually referred to as or . ${\ displaystyle U \ subseteq X}$ ${\ displaystyle s \ in \ Gamma (U, F)}$${\ displaystyle x \ in X}$${\ displaystyle s}$${\ displaystyle F_ {x}}$${\ displaystyle s}$${\ displaystyle \ mathrm {supp} \, s}$${\ displaystyle | s |}$

In particular, the term carrier of a vector field defined on a manifold denotes the termination of the set of points in which the vector field is not zero. ${\ displaystyle M}$ ${\ displaystyle F \ colon M \ to TM}$

The carrier of a cut is always completed by definition.

### Carrier of a sheaf

The bearer of itself is the set of points for which the stalk is non-zero. ${\ displaystyle F}$${\ displaystyle x \ in X}$${\ displaystyle F_ {x}}$

The carrier of a sheaf is not necessarily closed, but the carrier of a coherent module sheaf is .

## literature

• Roger Godement: Théorie des faisceaux . Hermann, Paris 1958.

## Individual evidence

1. The spelling may be confused with the trace of a square matrix, which is called trace in English .${\ displaystyle Tr (f)}$