homogeneous functionHomogeneousFunction2004-10-17 05:33:032008-06-08 13:48:00Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}\begin{defn} Suppose $V,\,W$ are a vector spaces over $\R$,
and $f\colon V \to W$ is a mapping.
\begin{itemize}
\item
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
a \PMlinkescapetext{\emph{homogeneous function of degree $r$}}.
\item
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
\PMlinkescapetext{\emph{absolutely homogeneous function of degree $r$}}.
\item
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 $f$ is
a \PMlinkescapetext{\emph{positively homogeneous function of degree $r$}}.
\end{itemize}
\end{defn}
\subsubsection*{Notes}
For any homogeneous function as above, $f(0)=0$.
When the \PMlinkescapetext{type} of homegeneity is clear one simply talks about
$r$-homogeneous functions.