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}}$

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

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