When concepts (properties, operationsMathworldPlanetmath, etc.) on a set Y are extended to functionsMathworldPlanetmath f:XY 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 sequenceMathworldPlanetmath {fn}n=1 of functions XY converges pointwise to a function f if limnfn(x)=f(x) for all xX.

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

(f+g)(x)= f(x)+g(x) (pointwise addition)
(fg)(x)= f(x)g(x) (pointwise multiplicationPlanetmathPlanetmath)
(λf)(x)= λf(x) (pointwise multiplication by scalar)

where the identitiesPlanetmathPlanetmathPlanetmath hold for all xX. 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