Editing Generalized function (section)

===Example: Colombeau algebra===

A simple example is obtained by using the polynomial scale on '''N''',
<math>s = \{ a_m:\mathbb N\to\mathbb R, n\mapsto n^m ;~ m\in\mathbb Z \}</math>. Then for any semi normed algebra (E,P), the factor space will be

:<math>G_s(E,P)= \frac{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\exists m\in\mathbb Z:p(f_n)=o(n^m)\}
}{
\{ f\in E^{\mathbb N}\mid\forall p\in P,\forall m\in\mathbb Z:p(f_n)=o(n^m)\}
}.</math>

In particular, for (''E'',&nbsp;''P'')=('''C''',|.|) one gets (Colombeau's) [[generalized number|generalized complex numbers]] (which can be "infinitely large" and "infinitesimally small" and still allow for rigorous arithmetics, very similar to [[non-standard analysis|nonstandard number]]s). For (''E'',&nbsp;''P'')&nbsp;=&nbsp;(''C<sup>∞</sup>''('''R'''),{''p<sub>k</sub>''}) (where  ''p<sub>k</sub>'' is the supremum of all derivatives of order less than or equal to ''k'' on the ball of radius ''k'') one gets [[Colombeau algebra|Colombeau's simplified algebra]].