pointwise

When concepts (properties, operations, etc.) on a set $Y$are extended to functions $f:X⟶Y$by treating each function value $f(x)$ in isolation, theextended concept is often qualified with the wordpointwise. One example is pointwise convergenceof functions—a sequence ${\{f_n\}}_{n=1}^{\mathrm{\infty }}$ offunctions $X⟶Y$ converges pointwise toa function $f$ if ${lim}_{n\to \mathrm{\infty }}{f}_n(x)=f(x)$for all $x\in X$.

An important of pointwise conceptsare the pointwise operations—operations definedon functions by applying the operations to function valuesseparately for each point in the domain of definition. Theseinclude

$(f+g)(x)=$$f(x)+g(x)$(pointwise addition)

$(f\cdot g)(x)=$$f(x)\cdot g(x)$(pointwise multiplication)

$(\lambda f)(x)=$$\lambda \cdot f(x)$(pointwise multiplication by scalar)

where the identities hold for all $x\in X$. Pointwiseoperations inherit such properties as associativity, commutativity,and distributivity from corresponding operations on $Y$.

An example of an operation on functions which is notpointwise is the convolution (http://planetmath.org/Convolution) product.