Properties of Evaluation Linear Transformation

Theorem

Let $R$ be a commutative ring.

Let $G$ be an $R$-module.

Let $G^*$ be the algebraic dual of $G$.

Let $\left \langle {x, t'} \right \rangle$ be the evaluation linear transformation from $G$ to $G^{**}$.

Then the mapping $\phi: G \times G^* \to R$ defined as $\forall \left({x, t'}\right) \in G \times G^*: \phi \left({x, t'}\right) = \left \langle {x, t'} \right \rangle$ satisfies the following properties:

$(1): \quad \forall x, y \in G: \forall t' \in G^*: \left \langle {x + y, t'} \right \rangle = \left \langle {x, t'} \right \rangle + \left \langle {y, t'} \right \rangle$
$(2): \quad \forall x \in G: \forall s', t' \in G^*: \left \langle {x, s' + t'} \right \rangle = \left \langle {x, s'} \right \rangle + \left \langle {x, t'} \right \rangle$
$(3): \quad \forall x \in G: \forall s', t' \in G^*: \forall \lambda \in R: \left \langle {\lambda x, t'} \right \rangle = \lambda \left \langle {x, t'} \right \rangle = \left \langle {x, \lambda t'} \right \rangle$