必要性:因为\(\langle\xi_1,\ldots,\xi_s\rangle\)是\(\phi\)-不变子空间且\(\xi_i\in\langle\xi_1,\ldots,\xi_s\rangle\),所以根据定义
\begin{equation*}
\phi(\xi_i)\in\langle\xi_1,\ldots,\xi_s\rangle,i=1,\ldots ,s.
\end{equation*}
充分性:对任意\(\alpha\in\langle\xi_1,\ldots,\xi_s\rangle\),存在\(a_1,\ldots,a_s\in\F\),使得
\begin{equation*}
\alpha=a_1\xi_1+\cdots+a_s\xi_s,
\end{equation*}
则
\begin{equation*}
\phi(\alpha)=a_1\phi(\xi_1)+\cdots+a_s\phi(\xi_s).
\end{equation*}
由于\(\phi(\xi_i)\in\langle\xi_1,\ldots,\xi_s\rangle,i=1,\ldots ,s\),所以
\begin{equation*}
a_1\phi(\xi_1)+\cdots+a_s\phi(\xi_s)\in\langle\xi_1,\ldots,\xi_s\rangle,
\end{equation*}
即\(\phi(\alpha)\in\langle\xi_1,\ldots,\xi_s\rangle\),由此推出\(\langle\xi_1,\ldots,\xi_s\rangle\)是\(\phi\)-不变子空间。