Homogeneous function: Difference between revisions

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==Glossary of name variants==

{{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,</math> <math>[0, \infty),</math> or <math>\RReals</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if

<math display=inline>f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in 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>

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The following commonly encountered special cases and variations of this definition have their own terminology:

#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}:{{sfn|Schechter|1996|pp=313-314}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>

#* ThisWhen the function <math>f</math> is valued in a vector space or field, then this property is often[[Logical alsoequivalence|logically calledequivalent]] to {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}} because for a function valued in a vector space or field, itwhich isby [[Logical equivalencedefinition{{sfn|Kubrusly|logically2011|p=200}} equivalent]] tomeans: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|non-negative}} real <math>r \geq 0.</math><ref group=proof>Assume that <math>f</math> is strictly positively homogeneous and valued in a vector space or a field. Then <math>f(0) = f(2 \cdot 0) = 2 f(0)</math> so subtracting <math>f(0)</math> from both sides shows that <math>f(0) = 0.</math> Writing <math>r := 0,</math> then for any <math>x \in X,</math> <math>f(r x) = f(0) = 0 = 0 f(x) = r f(x),</math> which shows that <math>f</math> is nonnegative homogeneous.</ref> However, for a functionfunctions valued in the [[extended real numbers]] <math>[-\infty, \infty] = \RReals \cup \{\pm \infty\},</math> which appear in fields like [[convex analysis]], the multiplication <math>0 \cdot f(x)</math> will be undefined whenever <math>f(x) = \pm \infty</math> and so these statements are not necessarily interchangeable.<ref group=note>However, if such an <math>f</math> satisfies <math>f(rx) = r f(x)</math> for all <math>r > 0</math> and <math>x \in X,</math> then necessarily <math>f(0) \in \{\pm \infty, 0\}</math> and whenever <math>f(0), f(x) \in \R</math> are both real then <math>f(r x) = r f(x)</math> will hold for all <math>r \geq 0.</math></ref>

#* This property is used in the definition of a [[sublinear function]].{{sfn|Schechter|1996|pp=313-314}}{{sfn|Kubrusly|2011|p=200}}

#* [[Minkowski functional]]s are exactly those non-negative extended real-valued functions with this property.

#{{em|{{visible anchor|Real homogeneity|Real homogeneous}}}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>