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

• Results about evaluation linear transformations can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations