Definition:Evaluation Linear Transformation/Module Theory
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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 $x^\wedge \in G^{**}$
- 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