Definitions: Let $V$ be a vector space and $M\subseteq V$ subspace. We define $V^* :=$ space of all linear transformations $f:V\to \mathbb{F}$ and $M^0:=\{f\in V^*| f(M)=0\}$.
In a class of (advanced) linear algebra my professor proposed, as exercise, to prove these two equalities: $$\ker(T^*) = (\text{Im}(T))^0,$$ and $$(\text{Im}(T^*))^0 = \ker(T),$$ for a linear function $T:V\to W.$ I'm OK with the first (I wrote it here to show that confusing both equalities is not the case). But the second I don't see how it can be true, once $\text{Im}(T^*)\subseteq V^*$ so $(\text{Im}(T^*))^0 \subseteq V^{**} := (V^*)^* $ and $\ker(T)\subseteq V$. Just wanted some confirmation if the equality is not (always) true.