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:


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

Proof

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


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