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 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.

单词	PartialMapping
释义	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 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.
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