# pointwise

When concepts (properties, operations, etc.) on a set $Y$ are extended to functions $f\colon X\longrightarrow Y$ by treating each function value $f(x)$ in isolation, the extended concept is often qualified with the word pointwise. One example is pointwise convergence of functions—a sequence $\{f_{n}\}_{n=1}^{\infty}$ of functions $X\longrightarrow Y$ converges pointwise to a function $f$ if $\lim_{n\rightarrow\infty}f_{n}(x)=f(x)$ for all $x\in X$.

An important of pointwise concepts are the pointwise operations—operations defined on functions by applying the operations to function values separately for each point in the domain of definition. These include

 $\displaystyle(f+g)(x)=$ $\displaystyle f(x)+g(x)$ (pointwise addition) $\displaystyle(f\cdot g)(x)=$ $\displaystyle f(x)\cdot g(x)$ (pointwise multiplication) $\displaystyle(\lambda f)(x)=$ $\displaystyle\lambda\cdot f(x)$ (pointwise multiplication by scalar)

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

An example of an operation on functions which is not pointwise is the convolution (http://planetmath.org/Convolution) product.

Title pointwise Pointwise 2013-03-22 15:25:00 2013-03-22 15:25:00 lars_h (9802) lars_h (9802) 4 lars_h (9802) Definition msc 03-00 msc 08-00 pointwise operation pointwise addition pointwise muliplication