Pointwise operation

In abstract algebra, pointwise operation is a way of extending an operation defined on an algebraic struture to a set of functions taking values in that structure.

If O is an n-ary operator on a set S, written in functional notation, and F is a set of functions from A to S, then the pointwise extension of O to F is the operator, also written O, defined on n-tuples of functions in F with value a function from A to S, as follows


 * $$O(f_1,\ldots,f_n) = ( x \mapsto O(f_1(x),\ldots,f_n(x)) ) .\,$$

In the common case of a binary operation $$\star$$, written now in operator notation, we can write


 * $$f \star g : x \mapsto f(x) \star g(x) .\,$$

For specific operations such as addition and multiplication the phrases "pointwise addition", "pointwise multiplication" are often used to denote their pointwise extension.