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==Glossary of name variants==
{{or section|date=December 2021}}
Let <math>X</math>f be: aX [[vector space]] over a field <math>\mathbb{F}</math>to and let <math>Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{G}</math> where <math>\mathbb{F}</math> and <math>\mathbb{G}</math> will (usually be (or possibly just contain as subsets) the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex.</math>). LetIf <math>f : X \to YS</math> beis a map.<refset group=note>Noteof inscalars, particularsuch that ifas <math>Y\Z, =[0, \Complex =infty), \mathbbtext{G},</math> thenor every} <math>\R</math>-valued functionfor onexample, then <math>Xf</math> is alsosaid to be {{em|{{visible anchor|homogeneous over}} <math>\ComplexS</math>-valued.</ref>}} if
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 '''{{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'''}} (resp.respectively, '''{{em|absolutely homogeneous'''}}) over this set if <math>f(s x) = \overline{s} f(x)</math> (resp.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>
The following commonly encountered special cases have their own terminology:<ref group=note>For a property such as real homogeneity to even be well-defined, the fields <math>\mathbb{F}</math> and <math>\mathbb{G}</math> must both contain the real numbers. We will of course automatically make whatever assumptions on <math>\mathbb{F}</math> and <math>\mathbb{G}</math> are necessary in order for the scalar products below to be well-defined.</ref>
#('''{{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}'''}}) '''{{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}'''}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
#* This property is often also called '''{{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}'''}} because for a function valued in a vector space or field, it is [[Logical equivalence|logically equivalent]] to: <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 function valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \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]].
#* [[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>
#* This property is used in the definition of a {{em|real}} [[linear functional]].
#'''{{em|{{visible anchor|Homogeneity|Homogeneous}}'''}}: <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* It is emphasized that this definition depends on the scalar field <math>\mathbb{F}</math> underlying the ___domain <math>X</math>.
#* This property is used in the definition of [[linear functional]]s and [[linear map]]s.
#'''{{em|[[Semilinear map|{{visible anchor|Conjugate homogeneity|Conjugate homogeneous}}]]'''}}: <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the [[complex conjugate]] of <math>s.</math> But more generally, as with [[semilinear map]]s for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math>
#* Along with [[Additive map|additivity]], this property is assumed in the definition of an [[antilinear map]]. It is also assumed that one of the two coordinates of a [[sesquilinear form]] has this property (such as the [[inner product]] of a [[Hilbert space]]).
All of the above definitions can be generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = |r| f(x),</math> in which case that definition is prefixed with the word {{nowrap|"'''{{em|absolute'''}}"}} or {{nowrap|"'''{{em|absolutely'''}}."}}
For example,
<ol start=5>
<li>'''{{em|{{visible anchor|Absolute homogeneity|Absolute homogeneous|Absolutely homogeneous}}'''}}: <math>f(sx) = |s| f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
* This property is used in the definition of a [[seminorm]] and a [[Norm (mathematics)|norm]].
</li>
</ol>
If <math>k</math> is a fixed real number then the above definitions can be further generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = r^k f(x)</math> (and similarly, by replacing <math>f(rx) = |r| f(x)</math> with <math>f(rx) = |r|^k f(x)</math> for conditions using the absolute value, etc.), in which case the homogeneity is said to be {{nowrap|"'''{{em|of degree <math>k</math>'''}}"}} (where in particular, all of the above definitions are {{nowrap|"'''{{em|of degree <math>1</math>'''}}"}}).
For instance,
<ol start=6>
<li>'''{{em|{{visible anchor|Real homogeneity of degree}} <math>k</math>'''}}: <math>f(rx) = r^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
</li>
<li>'''{{em|{{visible anchor|Homogeneity of degree}} <math>k</math>'''}}: <math>f(sx) = s^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
</li>
<li>'''{{em|{{visible anchor|Absolute real homogeneity of degree}} <math>k</math>'''}}: <math>f(rx) = |r|^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
</li>
<li>'''{{em|{{visible anchor|Absolute homogeneity of degree}} <math>k</math>'''}}: <math>f(sx) = |s|^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
</li>
</ol>
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