Carrier (mathematics)

This article covers the bearer of functions, distributions, cuts and sheaves. For the bearer of a measure see bearer (measure theory) .

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 . ${\displaystyle Tr(f)}$