Homogeneous function: Difference between revisions

Revision as of 19:04, 10 April 2023

Revision as of 19:07, 10 April 2023

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;'''Proofs'''

{{reflist|group=proof}}|refs=

<ref group=proof name=posHomEquivToNonnegHom>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>

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==References==