# partial mapping

Let ${X}_{1},\mathrm{\cdots},{X}_{n}$ and $Y$ be sets, and let $f$ be a function of $n$ variables: $f:{X}_{1}\times {X}_{2}\times \mathrm{\cdots}\times {X}_{n}\to Y$. ${x}_{i}\in {X}_{i}$ for $2\le i\le n$. The induced mapping $a\mapsto f(a,{x}_{2},\mathrm{\dots},{x}_{n})$ is called the *partial mapping* determined by $f$ corresponding to the first variable.

In the case where $n=2$, the map defined by $a\mapsto f(a,x)$ is often denoted $f(\cdot ,x)$. Further, any function $f:{X}_{1}\times {X}_{2}\to Y$ determines a mapping
from ${X}_{1}$ into the set of mappings of ${X}_{2}$ into $Y$, namely
$\overline{f}:x\mapsto (y\mapsto f(x,y))$.
The converse^{} holds too, and it is customary to identify $f$ with
$\overline{f}$. Many of the “canonical isomorphisms” that we come across (e.g. in multilinear algebra) are illustrations of this kind of identification.

Title | partial mapping |
---|---|

Canonical name | PartialMapping |

Date of creation | 2013-03-22 13:59:31 |

Last modified on | 2013-03-22 13:59:31 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 5 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 03E20 |