Pointwise operation

2020 Mathematics Subject Classification: Primary: 08-XX [MSN][ZBL]

Extension of an algebraic operation $\star$ on a set $X$ to a set of functions on a set $Y$ taking values in $X$. If $f, g$ are functions taking values in $X$ then the pointwise extension of a binary operation $\star$ is $$ f \star g : y \mapsto f(y) \star g(y)\,\ \ \text{for each}\ y \in Y \ . $$ The terms "pointwise addition", "pointwise multiplication" are also used. Algebraic operations of different signature have analogous pointwise extension.

This may be distinguished from such operations as convolution of functions, where the value of $f*g$ at $y$ does not depend solely on the values $f(y), g(y)$. See also Pointwise convergence.