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
Canonical name	Pointwise
Date of creation	2013-03-22 15:25:00
Last modified on	2013-03-22 15:25:00
Owner	lars_h (9802)
Last modified by	lars_h (9802)
Numerical id	4
Author	lars_h (9802)
Entry type	Definition
Classification	msc 03-00
Classification	msc 08-00
Defines	pointwise operation
Defines	pointwise addition
Defines	pointwise muliplication