# homogeneous function

###### Definition 1.

Suppose $V,\,W$ are a vector spaces over $\mathbbmss{R}$, and $f\colon V\to W$ is a mapping.

• If there exists an $r\in\mathbbmss{R}$, such that

 $f(\lambda v)=\lambda^{r}f(v)$

for all $\lambda\in\mathbbmss{R}$ and $v\in V$, then $f$ is a .

• If there exists an $r\in\mathbbmss{R}$, such that

 $f(\lambda v)=|\lambda|^{r}f(v)$

for all $\lambda\in\mathbbmss{R}$ and $v\in V$, then $f$ is .

• If there exists an $r\in\mathbbmss{R}$, such that

 $f(\lambda v)=\lambda^{r}f(v)$

for all $\lambda\geq 0$ and $v\in V$, then $f$ is a .

## Notes

For any homogeneous function as above, $f(0)=0$.

When the of homegeneity is clear one simply talks about $r$-homogeneous functions.

Title homogeneous function HomogeneousFunction 2013-03-22 14:44:37 2013-03-22 14:44:37 matte (1858) matte (1858) 8 matte (1858) Definition msc 15-00 positively homogeneous function of degree homogeneous function of degree positively homogeneous function HomogeneousPolynomial SubLinear