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 mappingfrom $X_1$ 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.