Underlying Mapping of Evaluation Linear Transformation is Element of Double Dual

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Let $\struct {R, +, \times}$ 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 $x \in G$, let $x^\wedge: G^* \to R$ be defined as:

$\forall t \in G^*: \map {x^\wedge} t = \map t x$


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


We have that $x^\wedge$ is a mapping from $G^* \to R$.

It remains to be demonstrates that $x^\wedge$ is in fact a linear transformation.

Hence we need to show that:

$(1): \quad \forall u, v \in G^*: \map {x^\wedge} {u + v} = \map {x^\wedge} u + \map {x^\wedge} v$
$(2): \quad \forall u \in G^*: \forall \lambda \in R: \map {x^\wedge} {\lambda \times u} = \lambda \times \map {x^\wedge} u$


\(\text {(1)}: \quad\) \(\ds \map {x^\wedge} {u + v}\) \(=\) \(\ds \map {\paren {u + v} } t\) Definition of $x^\wedge$
\(\ds \) \(=\) \(\ds \map u t + \map v t\) Definition of Pointwise Addition of Linear Transformations
\(\ds \) \(=\) \(\ds \map {x^\wedge} u + \map {x^\wedge} v\) Definition of $x^\wedge$


\(\text {(2)}: \quad\) \(\ds \map {x^\wedge} {\lambda \times u}\) \(=\) \(\ds \map {\paren {\lambda \times u} } t\) Definition of $x^\wedge$
\(\ds \) \(=\) \(\ds \lambda \times \map u t\) Definition of Linear Transformation
\(\ds \) \(=\) \(\ds \lambda \times \map {x^\wedge} u\) Definition of $x^\wedge$

Hence the result.


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