Evaluation Linear Transformation is Bilinear

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 as defined in Evaluation Linear Transformation.

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:


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


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


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