Editing Direct product (section)

== Direct product of modules ==
The direct product for [[module (mathematics)|modules]] (not to be confused with the [[tensor product of modules|tensor product]]) is very similar to the one that is defined for groups above by using the Cartesian product with the operation of addition being componentwise, and the [[scalar multiplication]] just distributing over all the components. Starting from <math>\R</math>, [[Euclidean space]] <math>\R^n</math> is gotten, the prototypical example of a real <math>n</math>-dimensional vector space. The direct product of <math>\R^m</math> and <math>\R^n</math> is <math>\R^{m+n}.</math>

A direct product for a finite index <math display="inline">\prod_{i=1}^n X_i</math> is canonically isomorphic to the [[direct sum of modules|direct sum]] <math display="inline">\bigoplus_{i=1}^n X_i.</math> The direct sum and the direct product are not isomorphic for infinite indices for which the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of [[category theory]]: the direct sum is the [[coproduct]], and the direct product is the product.

For example, for <math display="inline">X = \prod_{i=1}^\infty \R</math> and <math display="inline">Y = \bigoplus_{i=1}^\infty \R,</math> the infinite direct product and direct sum of the real numbers. Only sequences with a finite number of non-zero elements are in <math>Y.</math> For example, <math>(1, 0, 0, 0, \ldots)</math> is in <math>Y</math> but <math>(1, 1, 1, 1, \ldots)</math> is not. Both sequences are in the direct product <math>X;</math> in fact, <math>Y</math> is a proper subset of <math>X</math> (that is, <math>Y \subset X</math>).<ref>{{Cite web|url=http://mathworld.wolfram.com/DirectProduct.html|title=Direct Product| last = Weisstein | first = Eric W.|website=mathworld.wolfram.com|language=en|access-date=2018-02-10}}</ref><ref>{{Cite web |url=http://mathworld.wolfram.com/GroupDirectProduct.html|title=Group Direct Product| last = Weisstein | first = Eric W.| website=mathworld.wolfram.com | language=en|access-date=2018-02-10}}</ref>