Definition:Evaluation Linear Transformation/Module Theory

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

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

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

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

For each $x \in G$, we define the mapping $x^\wedge: G^* \to R$ as:
 * $\forall t \in G^*: \map {x^\wedge} t = \map t x$

The mapping $J: G \to G^{**}$ defined as:
 * $\forall x \in G: \map J x = x^\wedge$

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

It is usual to denote the mapping $t: G^* \to R$ as follows:


 * $\forall x \in G, t \in G^*: \innerprod x t := \map t x$

Also see

 * Underlying Mapping of Evaluation Linear Transformation is Element of Double Dual, demonstrating that $x^\wedge \in G^{**}$


 * Evaluation Linear Transformation is Linear Transformation