Evaluation Isomorphism is Isomorphism

Theorem
Let $R$ be a commutative ring with unity.

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 $\sequence {a_n}$ be an ordered basis of $G$.

Then $\sequence {\map J {a_n} }$ is the ordered basis of $G^{**}$ dual to the ordered basis of $G^*$ dual to $\sequence {a_n}$.

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

Also see

 * Definition:Evaluation Isomorphism: what an evaluation linear transformation is called when its dimension is finite.