Solution
= Solution
Suppose $U$ is an <invariant subspace> for $T$. For $\varphi\in U^0$ and $u\in U$, the <dual map> satisfies
$$
(T'\varphi)(u)=\varphi(Tu)=0,
$$
because $Tu\in U$ and the <annihilator of a vector subspace> consists of the <linear functionals> vanishing there. Thus
$$
\boxed{T'(U^0)\subseteq U^0}.
$$
This is the correspondence between <invariant subspaces and annihilators>.