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More generally, every [[norm (mathematics)|norm]] and [[seminorm]] is a positively homogeneous function of degree {{math|1}} which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.
===Linear
Any [[linear map]] <math>f : V \to W</math> between [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}} is homogeneous of degree 1, by the definition of linearity:
<math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math>
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