Pointwise operation

From Citizendium, the Citizens' Compendium

Jump to: navigation, search


This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
http://en.citizendium.org/wiki?title=Pointwise_operation&printable=yes
This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are subject to a disclaimer.

In abstract algebra, pointwise operation is a way of extending an operation defined on an algebraic struture to a set of functions taking values in that structure.

If O is an n-ary operator on a set S, written in functional notation, and F is a set of functions from A to S, then the pointwise extension of O to F is the operator, also written O, defined on n-tuples of functions in F with value a function from A to S, as follows

O(f_1,\ldots,f_n) = ( x \mapsto O(f_1(x),\ldots,f_n(x)) ) .\,

In the common case of a binary operation \star, written now in operator notation, we can write

f \star g : x \mapsto f(x) \star g(x) .\,

For specific operations such as addition and multiplication the phrases "pointwise addition", "pointwise multiplication" are often used to denote their pointwise extension.

Retrieved from "http://reid.citizendium.org/wiki/Pointwise_operation"
Views
Personal tools