Definition:Evaluation Isomorphism

Theorem
Let $R$ be a commutative ring.

Let $G$ be a unitary $R$-module whose dimension is finite.

Then the evaluation linear transformation $J: G \to G^{**}$ is an isomorphism.

Proof
Let $\left \langle {a_n} \right \rangle$ be an ordered basis of $G$.

Then $\left \langle {J \left({a_n}\right)} \right \rangle$ is the ordered basis of $G^{**}$ dual to the ordered basis of $G^*$ dual to $\left \langle {a_n} \right \rangle$.

From this it follows that $J$ is an isomorphism.