Definition:Evaluation Linear Transformation

Theorem
Let $R$ be a commutative ring.

Let $G$ be an $R$-module.

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

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

For each $x \in G$, we define the mapping $x^\wedge: G^* \to R$ as:
 * $\forall t' \in G^*: x^\wedge \left({t'}\right) = t' \left({x}\right)$

Then $x^\wedge \in G^{**}$.

The mapping $J: G \to G^{**}$ defined as:
 * $\forall x \in G: J \left({x}\right) = x^\wedge$

is a linear transformation.

This mapping $J$ is called the evaluation linear transformation from $G$ into $G^{**}$.

It is usual to denote the mapping $t': G^* \to G$ as follows:


 * $\forall x \in G, t' \in G^*: \left \langle {x, t'} \right \rangle := t' \left({x}\right)$

Proof

 * $x^\wedge \in G^{**}$:


 * $J: G \to G^{**}$ is a linear transformation: