# partial mapping

Let $X_{1},\cdots,X_{n}$ and $Y$ be sets, and let $f$ be a function of $n$ variables: $f:X_{1}\times X_{2}\times\cdots\times X_{n}\to Y$. $x_{i}\in X_{i}$ for $2\leq i\leq n$. The induced mapping $a\mapsto f(a,x_{2},\ldots,x_{n})$ 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 PartialMapping 2013-03-22 13:59:31 2013-03-22 13:59:31 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 03E20