Suppose that $V$ is a $\mathbb{F}$-vector space with some $\mathbb{F}$ a field, and denote $V^{'} := \{l: V \to \mathbb{F} \,\, \mid $ $l$ is linear$\}$, the dual space of $V$. Define the annihilator of a subset $S \subseteq V$ by
$$ S^{\perp} := \{l \in V^{'} \, \, \mid l(s)=0, \forall s \in S\} $$
It is true that if $S, T \subseteq V$ are subsets such that $S \subseteq T$, then $T^{\perp} \subseteq S^{\perp}$.
Now, assume $S^{\perp} \subseteq T^{\perp}$. On what conditions will make $T \subseteq S$ be true? If impossible, why?
Thanks.
EDIT
Is it when $S$ and $T$ are subspaces of each other? That is, is it when $S$ is a subspace of $T$ (assuming already that $S$ and $T$ are subspaces of $V$)?
EDIT 2
To make it clear, I want to know the conditions so that
If $S^{\perp} \subseteq T^{\perp}$, then $T \subseteq S$
is true.