Dimension of Preimage under Evaluation Isomorphism of Annihilator on Algebraic Dual

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

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

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

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

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

Then:
 * $\map {J^\gets} {N^\circ}$ is an $\paren {n - p}$-dimensional subspace of $G$

where

Proof
We have that $N$ is a $p$-dimensional subspace of $G^*$.

From Dimension of Annihilator on Algebraic Dual we have that $N^\circ$ has dimension $n - p$.

From Inverse Evaluation Isomorphism of Annihilator we also have that $J^{-1} \sqbrk {N^\circ}$ also has dimension $n - p$.