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 $\innerprod x {t'}$ be the evaluation linear transformation from $G$ to $G^{**}$.

Then the mapping $\phi: G \times G^* \to R$ defined as:
 * $\forall \tuple {x, t'} \in G \times G^*: \map \phi {x, t'} = \innerprod x {t'}$

satisfies the following properties:


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


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


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