View source for Pointwise operation

{{TEX|done}}{{MSC|08}}

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 operation]]s 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]].

See also [[Pointwise order]].