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.