Evaluation Linear Transformation is Bilinear

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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 bilinearity properties:


\(\text {(1)}: \quad\) \(\ds \forall x, y \in G: \forall t \in G^*: \, \) \(\ds \innerprod {x + y} t\) \(=\) \(\ds \innerprod x t + \innerprod y t\)
\(\text {(2)}: \quad\) \(\ds \forall x \in G: \forall s, t \in G^*: \, \) \(\ds \innerprod x {s + t}\) \(=\) \(\ds \innerprod x s + \innerprod x t\)
\(\text {(3)}: \quad\) \(\ds \forall x \in G: \forall t \in G^*: \forall \lambda \in R: \, \) \(\ds \innerprod {\lambda x} t\) \(=\) \(\ds \lambda \innerprod x t\)
\(\ds \) \(=\) \(\ds \innerprod x {\lambda t}\)


Proof

\(\text {(1)}: \quad\) \(\ds \forall x, y \in G: \forall t \in G^*: \, \) \(\ds \innerprod {x + y} t\) \(=\) \(\ds \map t {x + y}\) Definition of Evaluation Linear Transformation/Module Theory
\(\ds \) \(=\) \(\ds \map t x + \map t y\) $t$ is a Linear Transformation
\(\ds \) \(=\) \(\ds \innerprod x t + \innerprod y t\) Definition of Evaluation Linear Transformation/Module Theory


\(\text {(2)}: \quad\) \(\ds \forall x \in G: \forall s, t \in G^*: \, \) \(\ds \innerprod x {s + t}\) \(=\) \(\ds \map {\paren {s + t} } x\) Definition of Evaluation Linear Transformation/Module Theory
\(\ds \) \(=\) \(\ds \map s x + \map t x\) Definition of Pointwise Addition of Linear Transformations
\(\ds \) \(=\) \(\ds \innerprod x s + \innerprod x t\) Definition of Evaluation Linear Transformation/Module Theory


\(\text {(3)}: \quad\) \(\ds \forall x \in G: \forall t \in G^*: \forall \lambda \in R: \, \) \(\ds \innerprod {\lambda x} t\) \(=\) \(\ds \map t {\lambda x}\) Definition of Evaluation Linear Transformation/Module Theory
\(\ds \) \(=\) \(\ds \lambda \map t x\) $t$ is a Linear Transformation
\(\ds \) \(=\) \(\ds \lambda \innerprod x t\) Definition of Evaluation Linear Transformation/Module Theory


and:

\(\text {(3)}: \quad\) \(\ds \forall x \in G: \forall t \in G^*: \forall \lambda \in R: \, \) \(\ds \innerprod {\lambda x} t\) \(=\) \(\ds \map t {\lambda x}\) Definition of Evaluation Linear Transformation/Module Theory
\(\ds \) \(=\) \(\ds \map {\paren {\lambda t} } x\) $t$ is a Linear Transformation
\(\ds \) \(=\) \(\ds \innerprod x {\lambda t}\) Definition of Evaluation Linear Transformation/Module Theory

$\blacksquare$


Sources

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