Evaluation Linear Transformation is Linear Transformation

Theorem
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 $G^*$.

Let the mapping $J: G \to G^{**}$ be the evaluation linear transformation from $G$ into $G^{**}$ defined as:
 * $\forall x \in G: \map J x = x^\wedge$

where for each $x \in G$, $x^\wedge: G^* \to R$ is defined as:
 * $\forall t \in G^*: \map {x^\wedge} t = \map t x$

Then $J$ is a linear transformation.

Proof
From Underlying Mapping of Evaluation Linear Transformation is Element of Double Dual, we have that:


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

Hence $x^\wedge$ a fortiori is a linear transformation.

It remains to be shown that $J: G \to G^{**}$ is a linear transformation.

That is, that the following conditions are satisfied by $J$:


 * $(1): \quad \forall x, y \in G: \map J {x + y} = \map J x + \map J y$
 * $(2): \quad \forall x \in G: \forall \lambda \in R: \map J {\lambda \times x} = \lambda \times J x$

Hence:

and:

Hence the result.