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.