Homogeneous function: Difference between revisions

Revision as of 19:06, 18 January 2022

Revision as of 19:07, 18 January 2022

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{{or section|date=December 2021}}

Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex.</math>). If <math>S</math> is a set of scalars, such as <math>\Z, [0, \infty), \text{ or } \R</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if

<math display=block>f(s x) = s f(x) \qquad \text{ for every } x \in X \text{ and scalar } s \in S,.</math>

while it is called {{em|conjugate homogeneous}} (respectively, {{em|absolutely homogeneous}}) over this set if <math>f(s x) = \overline{s} f(x)</math> (respectively, if <math>f(s x) = |s| f(x)</math>) holds for all such <math>x \text{ and } s.</math>

For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R.</math>