homogeneous function

Definition 1.

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

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

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

  for all $\lambda \in ℝ$ and $v\in V$, then $f$ isa .

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

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

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

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

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

  for all $\lambda \ge 0$ and $v\in V$, then $f$ isa .

Notes

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

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