Results Concerning Annihilator of Vector Subspace

Theorem
Let $G$ be an $n$-dimensional vector space over a field.

Let $G^*$ be the algebraic dual of $G$.

Let $G^{**}$ be the algebraic dual of $G^*$.

Let $J: G \to G^{**}$ be the evaluation isomorphism.

Let $M$ be an $m$-dimensional subspace of $G$.

Let $N$ be a $p$-dimensional subspace of $G^*$.

Let $M^\circ$ be the annihilator of $M$.

Let $J^\gets: \powerset {G^{**} } \to \powerset G$ be the inverse image mapping of $J$.

Then the following results hold: