# Definition:Zero Mapping

## Definition

Let $\mathbb A$ be one of the standard number systems $\N,\Z,\Q,\R,\C$.

Let $S$ be a set.

Let $f_0: S \to \mathbb A$ denote the constant mapping:

$\forall x \in S: \map {f_0} x = 0$

Then $f_0$ is referred to as the zero mapping.

### Vector Space

Let $Y$ be a vector space.

Let $S$ be a set.

Let $\mathbf 0_Y$ be the identity element of $Y$.

Suppose $\mathbf 0 : S \to Y$ is a mapping such that:

$\forall x \in S: \map {\mathbf 0} x = \mathbf 0_Y$

Then $\mathbf 0$ is referred to as the zero mapping.

### Distribution

Let $\map \DD \R$ be the test function space.

Let $\mathbf 0 \in \map {\DD'} \R$ be a distribution.

Suppose:

$\forall \phi \in \map \DD \R : \map {\mathbf 0} \phi = 0$

Then $\mathbf 0$ is referred to as the zero distribution.

## Also known as

The zero mapping is often encountered in the context of real analysis, where $\mathbb A = \R$, in which case $f_0$ is referred to as the zero function.

It is often denoted $0: S \to R$:

$\forall x \in S: \map 0 s = 0$