\begin{definition}[Definition:Evaluation Linear Transformation/Module Theory]
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:
:$\forall x \in G, t \in G^*: \innerprod x t := \map t x$
\end{definition}